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I can find parallel and non-parallel missing segments to the trapezoid given the midpoints.

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I can find lengths of segments formed by tangent lines and secant lines with a circle.

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I can find measures of angles and arcs formed by tangent lines and secant lines in a circle.

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I can recognize the 2-d shape of a cross section of a 3-d shape.

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I can calculate the volume of pyramids, cones, and sphere solids.

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I can calculate the total surface area and lateral surface area of 3-d pyramids, cones, and spheres and use the area to find missing lengths of sides, heights, or slant heights.

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Permutations, Combinations, and Probability - Worksheet and Key

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Permutations, Combinations, and Probability - Worksheet and Key

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Permutations, Combinations, and Probability - Worksheet and Key

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Permutations vs Combinations - Worksheet and Key

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Permutations vs Combinations - Worksheet and Key

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Permutations vs Combinations - Worksheet and Key

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Permutations and Combinations - Worksheet and Key

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- So let's keep thinking about different ways to sit multiple people in the certain number of chairs. So let's say we have six people. We have person A, we have person B, we have person C, person D, person E, and we have person F. So we have six people. And for the sake of this video, we're going to say oh we want to figure out all the scenarios, all the possibilities, all the permutations, all the ways that we could put them into three chairs. So there's chair number one, chair number two and chair number three. This is all review. This is covered in the permutations video. But it'll be very instructive as we move into a new concept. So what are all the permutations of putting six different people into three chairs? Well, like we've seen before, we can start with the first chair. And we can say look if no one's sat-- If we haven't seated anyone yet, how many different people could we put in chair number one? Well there's six different people who could be in chair number one. Let me get a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. Now for each of those six scenarios, how many people, how many different people could sit in chair number two? Well each of those six scenarios we've taken one of the six people to sit in chair number one. So that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now if you want to say well what about for the three chairs? Well for each of these 30 scenarios, how many different people could you put in chair number three? Well you're still going to have four people standing up not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios, or your total number of permutations where we care who's sitting in which chair is six times five times four, which is equal to 120 permutations. Permutations. Now lemme, permutations. Now it's worth thinking about what permutations are counting. Now remember we care, when we're talking about permutations, we care about who's sitting in which chair. So for example, for example this is one permutation. And this would be counted as another permutation. And this would be counted as another permutation. This would be counted as another permutation. So notice these are all the same three people, but we're putting them in different chairs. And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that. So our thinking in the permutation world. We would count all of these. Or we would count this as six different permutations. These are going towards this 120. And of course we have other permutations where we would involve other people. Where we have, it could be F, B, C, F, C, B, F, A, C, F, F. Actually let me do it this way. I'm going to be a little bit more systematic. F, uh lemme do it. B, B, F, C, B, C, F. And obviously I could keep doing. I can do 120 of these. I'll do two more. You could have C, F, B. And then you could have C, B, F. So in the permutation world, these are, these are literally 12 of the 120 permutations. But what if we, what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting, or in which chair they're sitting. So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question. If I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video, and try to think of what that number would actually be. Well a big clue was when we essentially wrote all of the permutations where we've picked a group of three people. We see that there's six ways of arranging the three people. When you pick a certain group of three people, that turned into six permutations. And so if all you want to do is care about well how many different ways are there to choose three from the six? You would take your whole permutations. You would take your number of permutations. You would take your number of permutations. And then you would divide it by the number of ways to arrange three people. Number of ways to arrange, arrange three people. And we see that you can arrange three people, or even three letters. You can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. So there are 120 permutations here. If you said how many different arrangements are there of taking six people and putting them into three chairs? That's 120. But now we're asking another thing. We're saying if we start with 120 people, and we want to choose. Sorry if we're starting with six people and we want to figure out how many ways, how many combinations, how many ways are there for us to choose three of them? Then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination. It's the combination, A, B, C. I don't care what order they sit in. I have chosen them. I have chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination. It is F, C, and B. Once again I don't care about the order. I just care that I've chosen these three people. So how many ways are there to choose three people out of six? It is 20. It's the total number of permutations. It's 120 divided by the number of ways you can arrange three people.

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Intro to combinations (video) | Combinations | Khan Academy

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So let's keep thinking about different ways to sit multiple people in the certain number of chairs. So let's say we have six people. We have person A, we have person B, we have person C, person D, person E, and we have person F. So we have six people.

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- [Voiceover] We know that if we had five people, let's say person A, person B, person C, person D, and person E, and we wanted to put them in five different, let's say, positions, or chairs, so position one, position two, position three, position four, and position five. If we wanted to count the number of scenarios, or we could say the number of permutations of putting these five people in these five chairs, we could say, well, we have five different ... If we seated people in order, which we might as well do, we could say, look, five different people could sit in chair one. So, for each of those scenarios, four different people could sit in chair two. Now, for each of these scenarios, now, so we have 20 scenarios. Five times four, we have 20 scenarios where we've seated seat one and seat two, how many people could we now seat in seat three for each of those 20 scenarios? Well, three people haven't sat down yet, so there's three possibilities there. So now, there's five times four times three scenarios for seating the first three people. How many people are left for seat four? Well, two people haven't sat down yet, so there's two possibilities. So now there's five times four times three times two scenarios of seating the first four seats. For each of those, how many possibilities are there for the fifth seat? Well, one. For each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. So the number of permutations, the number of ... Let me write this down. The number of permutations, permutations, of seating these five people in five chairs is five factorial. Five factorial, which is equal to five times four times three times two times one, which, of course, is equal to, let's see, 20 times six, which is equal to 120. We have already covered this in a previous video. But now let's do something maybe more interesting, or maybe you might find it less interesting. Let's say that we still have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. Let's say that we only have three chairs. We have chair one, we have chair two, and we have chair three. How many ways can you have five people, where only three of them are going to sit down in these three chairs, and we care which chair they sit in? I encourage you to pause the video and think about it. I am assuming you have had your go at it. Let's use the same logic. If we seat them in order, and we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we had five different people, five different people could sit in seat one. For each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? In each of these scenarios, if one person has sat down, there's four people left who haven't been seated, so four people could sit in seat two. So we have five times four scenarios where we've seated seats one and seat two. For each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't seaten or sat three of the people yet, so for each of these 20, we could put three different people in seat three, so that gives us five times four times three scenarios. So this is equal to five times four times three scenarios, which is equal to, this is equal to 60. So there's 60 permutations of sitting five people in three chairs. Now this, and my brain, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, when we just did five different people in five different chairs, and we cared which seat they sit in, we had this five factorial. Factorial is kind of neat little operation there. How can I relate factorial to what we did just now? It looks like we kind of did factorial, but then we stopped. We stopped at, we didn't go times two times one. So one way to think about what we just did, is we just did five times four times three times two times one, but of course we actually didn't do the two times one, so you could take that and you could divide by two times one. If you did that, this two times one would cancel with that two times one and you'd be left with five times four times three. The whole reason I'm writing this way is that now I can write it in terms of factorial. I could write this as five factorial, five factorial, over two factorial, over two factorial. But then you might have the question where did this two come from? I have three seats. Where did this two come from? Well, think about it. I multiplied five times four times three, I kept going until I had that many seats, and then I didn't do the remainder. So the things that I left out, the things that I left out, that was essentially the number of people minus the number of chairs, so I was trying to put five things in three places. Five minus three, that gave me two left over. So I could write it like this. I could write it as five ... Let me use the same colors. I could write it as five factorial over, over five minus three, which of course is two, five minus three factorial. Another way of thinking about it, if we wanted to generalize, is if you're trying to figure out the number of permutations and there's a bunch of notations for writing this, if you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations you could put n people in r seats, and there's other notations as well, well, this is just going to be n factorial over n minus r factorial. Here n was five, r was three. Five minus three is two. Now, you'll see this in a probability or a statistics class, and people might memorize this thing. It seems like this kind of daunting thing. I'll just tell you right now, the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook, or what you might see in a class, or when you see this type of formula, you see that it's not some type of voodoo magic. But I will tell that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going, wait, does this formula apply there? What's n? What's r? But if you reason it through, it comes out of straight logic. You don't have to memorize anything. You don't feel like you're just memorizing without understanding. You're just using your deductive reasoning, your logic. That's especially valuable because as we'll see, not every scenario's going fit so cleanly into what we did. There might be some tweaks on this, where like maybe only person B likes sitting in one of the chairs, or who knows what it might be? Then your formula is going to be useless. So I like reasoning through it like this, but I just showed you this so that you could connect to it a formula that you might see in a lecture or in a class.

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Permutation formula (video) | Permutations | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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What's the Difference?

In English we use the word "combination" loosely, without thinking if the order of things is important. In other words:

"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

"The combination to the safe is 472". Now we do care about the order. "724" won't work, nor will "247". It has to be exactly 4-7-2.

So, in Mathematics we use more precise language:

When the order doesn't matter, it is a Combination.

When the order does matter it is a Permutation.

So, we should really call this a "Permutation Lock"!

In other words:

A Permutation is an ordered Combination.

To help you to remember, think " Permutation ... P osition"

Permutations

There are basically two types of permutation:

1. Repetition is Allowed: such as the lock above. It could be "333".

2. No Repetition: for example the first three people in a running race. You can't be first and second.

1. Permutations with Repetition

These are the easiest to calculate.

When a thing has n different types ... we have n choices each time!

For example: choosing 3 of those things, the permutations are:

n × n × n (n multiplied 3 times)

More generally: choosing r of something that has n different types, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)

Which is easier to write down using an exponent of r:
n × n × ... (r times) = nr
Example: in the lock above, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them:
10 × 10 × ... (3 times) = 10³ = 1,000 permutations
So, the formula is simply:

2. Permutations without Repetition

In this case, we have to reduce the number of available choices each time.

Example: what order could 16 pool balls be in?

After choosing, say, number "14" we can't choose it again.

So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, 12, 11, ... etc. And the total permutations are:
16 × 15 × 14 × 13 × ... = 20,922,789,888,000
But maybe we don't want to choose them all, just 3 of them, and that is then:
16 × 15 × 14 = 3,360
In other words, there are 3,360 different ways that 3 pool balls could be arranged out of 16 balls.
Without repetition our choices get reduced each time.
But how do we write that mathematically? Answer: we use the " factorial function"

The factorial function (symbol: ) just means to multiply a series of descending natural numbers. Examples:

4! = 4 × 3 × 2 × 1 = 24

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

1! = 1

Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.

So, when we want to select all of the billiard balls the permutations are:
16! = 20,922,789,888,000
But when we want to select just 3 we don't want to multiply after 14. How do we do that? There is a neat trick: we divide by 13!

That was neat: the 13 × 12 × ... etc gets "cancelled out", leaving only 16 × 15 × 14.

The formula is written:

Example Our "order of 3 out of 16 pool balls example" is:

(which is just the same as: 16 × 15 × 14 = 3,360)

Example: How many ways can first and second place be awarded to 10 people?

(which is just the same as: 10 × 9 = 90)

Notation

Instead of writing the whole formula, people use different notations such as these:

Examples:

Combinations

There are also two types of combinations (remember the order does not matter now):

1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)

2. No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition

Actually, these are the hardest to explain, so we will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!

The easiest way to explain it is to:

assume that the order does matter (ie permutations),

then alter it so the order does not matter.

Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order.

We already know that 3 out of 16 gave us 3,360 permutations.

But many of those are the same to us now, because we don't care what order!

For example, let us say balls 1, 2 and 3 are chosen. These are the possibilites:

So, the permutations have 6 times as many possibilites.

In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:

(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)

So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more):

That formula is so important it is often just written in big parentheses like this:

It is often called "n choose r" (such as "16 choose 3")

And is also known as the Binomial Coefficient.

Notation

All these notations mean "n choose r":

Just remember the formula:

Example: Pool Balls (without order)

So, our pool ball example (now without order) is:

= 560

So choosing 3 balls out of 16, or choosing 13 balls out of 16, have the same number of combinations:

In fact the formula is nice and symmetrical:

Also, knowing that 16!/13! reduces to 16×15×14, we can save lots of calculation by doing it this way:

= 560

Pascal's Triangle

We can also use Pascal's Triangle to find the values. Go down to row "n" (the top row is 0), and then along "r" places and the value there is our answer. Here is an extract showing row 16:

1 14 91 364 ...
1 15 105 455 1365 ...
1 16 120 560 1820 4368 ...

1. Combinations with Repetition

OK, now we can tackle this one ...

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla.

We can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections include

{c, c, c} (3 scoops of chocolate)

{b, l, v} (one each of banana, lemon and vanilla)

{b, v, v} (one of banana, two of vanilla)

(And just to be clear: There are n=5 things to choose from, we choose r=3 of them,
order does not matter, and we can repeat!)

Now, I can't describe directly to you how to calculate this, but I can show you a special technique that lets you work it out.

Think about the ice cream being in boxes, we could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and we will have 3 scoops of chocolate!

So it is like we are ordering a robot to get our ice cream, but it doesn't change anything, we still get what we want.

We can write this down as (arrow means move, circle means scoop).

In fact the three examples above can be written like this:

{c, c, c} (3 scoops of chocolate):

{b, l, v} (one each of banana, lemon and vanilla):

{b, v, v} (one of banana, two of vanilla):

So instead of worrying about different flavors, we have a simpler question: "how many different ways can we arrange arrows and circles?"

Notice that there are always 3 circles (3 scoops of ice cream) and 4 arrows (we need to move 4 times to go from the 1st to 5th container).

So (being general here) there are r + (n−1) positions, and we want to choose r of them to have circles.

This is like saying "we have r + (n−1) pool balls and want to choose r of them". In other words it is now like the pool balls question, but with slightly changed numbers. And we can write it like this:

Interestingly, we can look at the arrows instead of the circles, and say "we have r + (n−1) positions and want to choose (n−1) of them to have arrows", and the answer is the same:

So, what about our example, what is the answer?

There are 35 ways of having 3 scoops from five flavors of icecream.

In Conclusion

Phew, that was a lot to absorb, so maybe you could read it again to be sure!

But knowing how these formulas work is only half the battle. Figuring out how to interpret a real world situation can be quite hard.

But at least you now know the 4 variations of "Order does/does not matter" and "Repeats are/are not allowed":

708, 1482, 709, 1483, 747, 1484, 748, 749, 1485, 750

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In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

# data[].attributes.attachment_link.readable_text
What's the Difference?

In English we use the word "combination" loosely, without thinking if the order of things is important. In other words:

"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

"The combination to the safe is 472". Now we do care about the order. "724" won't work, nor will "247". It has to be exactly 4-7-2.

So, in Mathematics we use more precise language:

When the order doesn't matter, it is a Combination.

When the order does matter it is a Permutation.

So, we should really call this a "Permutation Lock"!

In other words:

A Permutation is an ordered Combination.

To help you to remember, think " Permutation ... P osition"

Permutations

There are basically two types of permutation:

1. Repetition is Allowed: such as the lock above. It could be "333".

2. No Repetition: for example the first three people in a running race. You can't be first and second.

1. Permutations with Repetition

These are the easiest to calculate.

When a thing has n different types ... we have n choices each time!

For example: choosing 3 of those things, the permutations are:

n × n × n (n multiplied 3 times)

More generally: choosing r of something that has n different types, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)

Which is easier to write down using an exponent of r:
n × n × ... (r times) = nr
Example: in the lock above, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them:
10 × 10 × ... (3 times) = 10³ = 1,000 permutations
So, the formula is simply:

2. Permutations without Repetition

In this case, we have to reduce the number of available choices each time.

Example: what order could 16 pool balls be in?

After choosing, say, number "14" we can't choose it again.

So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, 12, 11, ... etc. And the total permutations are:
16 × 15 × 14 × 13 × ... = 20,922,789,888,000
But maybe we don't want to choose them all, just 3 of them, and that is then:
16 × 15 × 14 = 3,360
In other words, there are 3,360 different ways that 3 pool balls could be arranged out of 16 balls.
Without repetition our choices get reduced each time.
But how do we write that mathematically? Answer: we use the " factorial function"

The factorial function (symbol: ) just means to multiply a series of descending natural numbers. Examples:

4! = 4 × 3 × 2 × 1 = 24

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

1! = 1

Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.

So, when we want to select all of the billiard balls the permutations are:
16! = 20,922,789,888,000
But when we want to select just 3 we don't want to multiply after 14. How do we do that? There is a neat trick: we divide by 13!

That was neat: the 13 × 12 × ... etc gets "cancelled out", leaving only 16 × 15 × 14.

The formula is written:

Example Our "order of 3 out of 16 pool balls example" is:

(which is just the same as: 16 × 15 × 14 = 3,360)

Example: How many ways can first and second place be awarded to 10 people?

(which is just the same as: 10 × 9 = 90)

Notation

Instead of writing the whole formula, people use different notations such as these:

Examples:

Combinations

There are also two types of combinations (remember the order does not matter now):

1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)

2. No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition

Actually, these are the hardest to explain, so we will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!

The easiest way to explain it is to:

assume that the order does matter (ie permutations),

then alter it so the order does not matter.

Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order.

We already know that 3 out of 16 gave us 3,360 permutations.

But many of those are the same to us now, because we don't care what order!

For example, let us say balls 1, 2 and 3 are chosen. These are the possibilites:

So, the permutations have 6 times as many possibilites.

In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:

(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)

So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more):

That formula is so important it is often just written in big parentheses like this:

It is often called "n choose r" (such as "16 choose 3")

And is also known as the Binomial Coefficient.

Notation

All these notations mean "n choose r":

Just remember the formula:

Example: Pool Balls (without order)

So, our pool ball example (now without order) is:

= 560

So choosing 3 balls out of 16, or choosing 13 balls out of 16, have the same number of combinations:

In fact the formula is nice and symmetrical:

Also, knowing that 16!/13! reduces to 16×15×14, we can save lots of calculation by doing it this way:

= 560

Pascal's Triangle

We can also use Pascal's Triangle to find the values. Go down to row "n" (the top row is 0), and then along "r" places and the value there is our answer. Here is an extract showing row 16:

1 14 91 364 ...
1 15 105 455 1365 ...
1 16 120 560 1820 4368 ...

1. Combinations with Repetition

OK, now we can tackle this one ...

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla.

We can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections include

{c, c, c} (3 scoops of chocolate)

{b, l, v} (one each of banana, lemon and vanilla)

{b, v, v} (one of banana, two of vanilla)

(And just to be clear: There are n=5 things to choose from, we choose r=3 of them,
order does not matter, and we can repeat!)

Now, I can't describe directly to you how to calculate this, but I can show you a special technique that lets you work it out.

Think about the ice cream being in boxes, we could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and we will have 3 scoops of chocolate!

So it is like we are ordering a robot to get our ice cream, but it doesn't change anything, we still get what we want.

We can write this down as (arrow means move, circle means scoop).

In fact the three examples above can be written like this:

{c, c, c} (3 scoops of chocolate):

{b, l, v} (one each of banana, lemon and vanilla):

{b, v, v} (one of banana, two of vanilla):

So instead of worrying about different flavors, we have a simpler question: "how many different ways can we arrange arrows and circles?"

Notice that there are always 3 circles (3 scoops of ice cream) and 4 arrows (we need to move 4 times to go from the 1st to 5th container).

So (being general here) there are r + (n−1) positions, and we want to choose r of them to have circles.

This is like saying "we have r + (n−1) pool balls and want to choose r of them". In other words it is now like the pool balls question, but with slightly changed numbers. And we can write it like this:

Interestingly, we can look at the arrows instead of the circles, and say "we have r + (n−1) positions and want to choose (n−1) of them to have arrows", and the answer is the same:

So, what about our example, what is the answer?

There are 35 ways of having 3 scoops from five flavors of icecream.

In Conclusion

Phew, that was a lot to absorb, so maybe you could read it again to be sure!

But knowing how these formulas work is only half the battle. Figuring out how to interpret a real world situation can be quite hard.

But at least you now know the 4 variations of "Order does/does not matter" and "Repeats are/are not allowed":

708, 1482, 709, 1483, 747, 1484, 748, 749, 1485, 750

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In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

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The Fundamental Counting Principle - Worksheet and Key

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Sample Spaces and The Counting Principle - Worksheet and Key

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For example, consider the number of ways in which a set of five different books can be arranged on a shelf. Since there are five books, there are five decisions. One must select the first book, then the second, the third, the fourth, and finally the fifth.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Definition, Examples, and Practice Problems

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Definition, Examples, and Practice Problems

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This is an interview with Dr. Jans Aasman, CEO of Franz, Inc. and designer of the Allegrograph knowledge graph engine. In this interview, we cover everything from the role of Lisp (and Lispers), the versatility of RDF hypergraphs, the value of Allegrograph, and the future of artificial intelligence, machine learning and inferential logic in the graph space.

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DataScienceCentral.com - Big Data News and Analysis

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This is an interview with Dr. Jans Aasman, CEO of Franz, Inc. and designer of the Allegrograph knowledge graph engine. In this interview, we cover everything from the role of Lisp (and Lispers), the versatility of RDF hypergraphs, the value of Allegrograph, and the future of artificial intelligence, machine learning and inferential logic in the graph space.

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Conditional Probability - Worksheet and Key

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Probability - Conditional and Two-Way Tables (Worksheet)

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Sal solves a conditional probability example where he thinks about probabilities like P(A | B) where the events are about lunch and breakfast!

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Proof: Radius is perpendicular to tangent line

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Sal proves that the radius that connects the intersection point of a tangent line with the circle is perpendicular to the tangent line.

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Formulas for Angles in Circles - MathBitsNotebook(Geo - CCSS Math)

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MathBitsNotebook Geometry CCSS Lessons and Practice is a free site for students (and teachers) studying high school level geometry under the Common Core State Standards.

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Conjectures in Geometry: Inscribed Angles

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The precise statements of the conjectures are given below. Each conjecture has a linked Sketch Pad demonstration to illustrate its truth (proof by Geometer's Sketch Pad!). The linked activities sheet also include directions for further "hands on" investigations involving these conjectures, as well as geometric problems which utilize their results.

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Central angle of a circle - Math Open Reference

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Definition and properties of the central angle of a circle

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Major and Minor Arcs - Math Open Reference

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Difference between major and minor arcs. How they can be ambiguous.

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Arc Length and Sector Area - Worksheet and Key

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Arc Length and Sector Area - Worksheet and Key

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Arc Length and Sector Area - Worksheet and Key

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I can use combinations (order does not matter) to calculate the number of ways for an arrangement of items.

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I can use permutations (order matters) to calculate the number of ways for an arrangement of items.

# data[].attributes.body
I can use the Fundamental Counting Principle for a very large sample space to model and calculate the number of ways something occurs.

# data[].attributes.body
I can use area models and two-way tables for calculating conditional probabilities.

# data[].attributes.body
I can use relationships between lines/segments and angles in circles to solve problems (including tangent and secant lines).

# data[].attributes.body
I can graph a circle from center and radius and write the equation. I can use relationships between angles and arcs in circles to solve problems.

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Only complete the second page (#

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Arc Length and Sector Area - Worksheet and Key

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A circle with area 81 pi has a sector with a 350-degree central angle. So this whole sector right over here that's shaded in, this pale orange-yellowish color, that has a 350-degree central angle. So you see the central angle, it's a very large angle. It's going all the way around like that. And they ask us, what is the area of the sector? So we just need to realize that the ratio between the area of the sector and the total area of the circle. And they tell us what the total area is. It's 81 pi. And 81 pi is going to be equal to the ratio of its central angle, which is 350 degrees, over the total number of degrees in a circle-- over 360. So the area of the sector over the total area is equal to the degrees in the central angle over the total degrees in a circle. And then we just can solve for area of a sector by multiplying both sides by 81 pi. 81 pi, 81 pi-- so these cancel out. 350 divided by 360 is 35/36. And so our area, our sector area, is equal to-- let's see, in the numerator, we have 35 times-- instead of 81, let's see, that's going to be 9 times 9 pi. And in the denominator, I have 36. Well, that's the same thing as 9 times 4. And so we can divide the numerator and the denominator both by 9, and so we are left with 35 times 9. And neither of these are divisible by 4, so that's about as simplified as we can get it. So let's think about what 35 times 9 is. 35 times 9, it's going to be 350 minus 35, which would be 315, I guess. Did I do that right? Yeah, it's going to be 270 plus 45, which is 315 pi over 4. 315 pi over 4 is the area of the sector.

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Area of a sector (video) | Sectors | Khan Academy

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A circle with area 81 pi has a sector with a 350-degree central angle. So this whole sector right over here that's shaded in, this pale orange-yellowish color, that has a 350-degree central angle. So you see the central angle, it's a very large angle. It's going all the way around like that.

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Area of a sector of a circle - Math Open Reference

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The formula used to find the area of a circlular sector - a pie-shaped part of a circle.

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Measurements in Similar Figures - MathBitsNotebook(Geo - CCSS Math)

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MathBitsNotebook Geometry CCSS Lessons and Practice is a free site for students (and teachers) studying high school level geometry under the Common Core State Standards.

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Surface area using a net: rectangular prism

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A polyhedron is a three-dimensional shape that has flat surfaces and straight edges. Learn whether or not a certain net could be folded up into a certain rectangular prism.

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Select the cube, face, or segment along the left navigation.

Then, place the object on the grid where you want it. If your selection is red, on the grid, then it is a location where you can not place the object.

HINT

Draw your shape from back to front and from bottom to top, to assure proper alignment of cubes.

When adding adjacent cubes, be sure to click on the face of the cube you want to be touching.

Create Mode

There are two ways to move objects:

1. Using the Arrow. Simply select and drag the object(s) to a new location.

2. Unit Movement Buttons

The other buttons along the top navigation serve various functions:

Rotate the entire figure by dragging the image or by using the sliders.

Use the Eraser to erase individual objects. You can also select multiple objects with the Pointer and then use the eraser to erase those objects.

Explode- Change all cubes into faces. If two cubes share a face, the face will not be shown. Note that you can not undo this action.

Use the Paint Brush to select a color before you create an object. You can also change the color once an object is created by selecting the object with the color you wish it to be.

Inspect Mode

In the Inspect Mode, you can:

See a 3D or 2D version of your isometric drawing.

Print your isometric drawing, as it is shown in the workspace.

Rotate your isometric drawing.

Choose to display your isometric drawing with or without axis.

Choose to display a solid or transparent isometric drawing.

You can reset all the view at any point by choosing the Reset View button.

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Use this interactive tool to create dynamic drawings on isometric dot paper.

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If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

700 possible mastery points

About this unit

Volume and surface area help us measure the size of 3D objects. We'll start with the volume and surface area of rectangular prisms. From there, we'll tackle trickier objects, such as cones and spheres.

Level up on the above skills and collect up to 320 Mastery points Start quiz

Level up on the above skills and collect up to 240 Mastery points Start quiz

Level up on all the skills in this unit and collect up to 700 Mastery points! Start Unit test

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Volume | Basic geometry and measurement | Math | Khan Academy

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Volume and surface area help us measure the size of 3D objects. We'll start with the volume and surface area of rectangular prisms. From there, we'll tackle trickier objects, such as cones and spheres.

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Surface Area of Prisms and Cylinders - Worksheet and Key

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The surface area is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid.

The volume is a measure of how much a figure can hold and is measured in cubic units. The volume tells us something about the capacity of a figure.

A prism is a solid figure that has two parallel congruent sides that are called bases that are connected by the lateral faces that are parallelograms. There are both rectangular and triangular prisms.

To find the surface area of a prism (or any other geometric solid) we open the solid like a carton box and flatten it out to find all included geometric forms.

To find the volume of a prism (it doesn't matter if it is rectangular or triangular) we multiply the area of the base, called the base area B, by the height h.

$$V=B\cdot h$$

A cylinder is a tube and is composed of two parallel congruent circles and a rectangle which base is the circumference of the circle.
Example
The area of one circle is:

$$A=\pi r^{2}$$

$$A=\pi \cdot 2^{2}$$

$$A=\pi \cdot 4$$

$$A\approx 12.6$$

The circumference of a circle:

$$C=\pi d$$

$$C=\pi \cdot 4$$

$$C\approx 12.6$$

The area of the rectangle:

$$A=C\cdot h$$

$$A=12.6 \cdot 6$$

$$A\approx 75.6$$

The surface area of the whole cylinder:

$$A=75.6+12.6+12.6=100.8\, units^{2}$$

To find the volume of a cylinder we multiply the base area (which is a circle) and the height h.

$$V=\pi r^{2}\cdot h$$

A pyramid consists of three or four triangular lateral surfaces and a three or four sided surface, respectively, at its base. When we calculate the surface area of the pyramid below we take the sum of the areas of the 4 triangles area and the base square. The height of a triangle within a pyramid is called the slant height.

The volume of a pyramid is one third of the volume of a prism.

$$V=\frac{1}{3}\cdot B\cdot h$$

The base of a cone is a circle and that is easy to see. The lateral surface of a cone is a parallelogram with a base that is half the circumference of the cone and with the slant height as the height. This can be a little bit trickier to see, but if you cut the lateral surface of the cone into sections and lay them next to each other it's easily seen.

The surface area of a cone is thus the sum of the areas of the base and the lateral surface:

$$A_{base}=\pi r^{2}\: and\: A_{LS}=\pi rl$$

$$A=\pi r^{2}+\pi rl$$
Example
$$\begin{matrix} A_{base}=\pi r^{2}\: \: &\, \, and\, \, & A_{LS}=\pi rl\: \: \: \: \: \: \: \\ A_{base}=\pi \cdot 3^{2} & & A_{LS}=\pi \cdot 3\cdot 9\\ A_{base}\approx 28.3\: \: && A_{LS}\approx 84.8\: \: \: \: \: \\ \end{matrix}$$

$$A=\pi r^{2}+\pi rl=28.3+84.8=113.1\, units^{2}$$

The volume of a cone is one third of the volume of a cylinder.

$$V=\frac{1}{3}\pi \cdot r^{2}\cdot h$$
Example
Find the volume of a prism that has the base 5 and the height 3.

$$B=3\cdot 5=15$$

$$V=15\cdot 3=45\: units^{3}$$

Video lesson

Find the surface area of a cylinder with the radius 4 and height 8

Find the volume of a cone with height 5 and the radius 3

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The surface area and the volume of pyramids, prisms, cylinders and cones (Geometry, Area) - Mathplanet

# data[].attributes.attachment_link.description
The surface area is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid.

# data[].attributes.attachment_link.readable_text
The surface area is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid.

The volume is a measure of how much a figure can hold and is measured in cubic units. The volume tells us something about the capacity of a figure.

A prism is a solid figure that has two parallel congruent sides that are called bases that are connected by the lateral faces that are parallelograms. There are both rectangular and triangular prisms.

To find the surface area of a prism (or any other geometric solid) we open the solid like a carton box and flatten it out to find all included geometric forms.

To find the volume of a prism (it doesn't matter if it is rectangular or triangular) we multiply the area of the base, called the base area B, by the height h.

$$V=B\cdot h$$

A cylinder is a tube and is composed of two parallel congruent circles and a rectangle which base is the circumference of the circle.
Example
The area of one circle is:

$$A=\pi r^{2}$$

$$A=\pi \cdot 2^{2}$$

$$A=\pi \cdot 4$$

$$A\approx 12.6$$

The circumference of a circle:

$$C=\pi d$$

$$C=\pi \cdot 4$$

$$C\approx 12.6$$

The area of the rectangle:

$$A=C\cdot h$$

$$A=12.6 \cdot 6$$

$$A\approx 75.6$$

The surface area of the whole cylinder:

$$A=75.6+12.6+12.6=100.8\, units^{2}$$

To find the volume of a cylinder we multiply the base area (which is a circle) and the height h.

$$V=\pi r^{2}\cdot h$$

A pyramid consists of three or four triangular lateral surfaces and a three or four sided surface, respectively, at its base. When we calculate the surface area of the pyramid below we take the sum of the areas of the 4 triangles area and the base square. The height of a triangle within a pyramid is called the slant height.

The volume of a pyramid is one third of the volume of a prism.

$$V=\frac{1}{3}\cdot B\cdot h$$

The base of a cone is a circle and that is easy to see. The lateral surface of a cone is a parallelogram with a base that is half the circumference of the cone and with the slant height as the height. This can be a little bit trickier to see, but if you cut the lateral surface of the cone into sections and lay them next to each other it's easily seen.

The surface area of a cone is thus the sum of the areas of the base and the lateral surface:

$$A_{base}=\pi r^{2}\: and\: A_{LS}=\pi rl$$

$$A=\pi r^{2}+\pi rl$$
Example
$$\begin{matrix} A_{base}=\pi r^{2}\: \: &\, \, and\, \, & A_{LS}=\pi rl\: \: \: \: \: \: \: \\ A_{base}=\pi \cdot 3^{2} & & A_{LS}=\pi \cdot 3\cdot 9\\ A_{base}\approx 28.3\: \: && A_{LS}\approx 84.8\: \: \: \: \: \\ \end{matrix}$$

$$A=\pi r^{2}+\pi rl=28.3+84.8=113.1\, units^{2}$$

The volume of a cone is one third of the volume of a cylinder.

$$V=\frac{1}{3}\pi \cdot r^{2}\cdot h$$
Example
Find the volume of a prism that has the base 5 and the height 3.

$$B=3\cdot 5=15$$

$$V=15\cdot 3=45\: units^{3}$$

Video lesson

Find the surface area of a cylinder with the radius 4 and height 8

Find the volume of a cone with height 5 and the radius 3

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The surface area and the volume of pyramids, prisms, cylinders and cones (Geometry, Area) - Mathplanet

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The surface area is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid.

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Surface Area Formulas

(Math | Geometry | Surface Area Formulas)

(pi = = 3.141592...)

Surface Area Formulas In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Cube | Rectangular Prism | Prism | Sphere | Cylinder | Units

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

Surface Area of a Cube = 6 a 2

(a is the length of the side of each edge of the cube)

In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

(a, b, and c are the lengths of the 3 sides)

In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(b is the shape of the ends)

Surface Area = Lateral area + Area of two ends

(r is radius of circle)

(Lateral area) = (perimeter of shape b) * L

(h is the height of the cylinder, r is the radius of the top)

Surface Area = Areas of top and bottom +Area of the side

These equations will give you correct answers if you keep the units straight. For example - to find the surface area of a cube with sides of 5 inches, the equation is:

Surface Area = 6*(5 inches) ²

= 6*(25 square inches)

= 150 sq. inches

Surface Area = 2(Area of top) + (perimeter of top)* height

Surface Area = 2( pi r 2) + (2 pi r)* h

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.

You can find the area of the top (or the bottom). That's the formula for area of a circle ( pi r ²). Since there is both a top and a bottom, that gets multiplied by two.

The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.

Add those two parts together and you have the formula for the surface area of a cylinder.

Surface Area = 2( pi r 2) + (2 pi r)* h

Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.

Surface Area of Any Prism

Surface Area of a Sphere = 4 pi r 2

Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h

Tip! Don't forget the units.

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Surface Area Formulas (Math | Geometry | Surface Area Formulas) (pi = = 3.141592...) Surface Area Formulas In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.

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If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

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Intro to nets of polyhedra (video) | Khan Academy

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Admittedly, "nets of polyhedra" sounds like the title of a bad sci-fi movie about man-eating, multi-headed fish. But in reality, nets of polyhedra are just 2D objects that wrap around 3D objects, like wrapping paper around presents.

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I can find an area of a sector of a circle given the central angle and its radius.

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I can relate similar side/perimeter to r, area of similar figures by r², and volume of similar figures by r³. I can use this concept to work between all of the ratios (r, r², and r³).

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I can use a mat plan/net to find dimensions, draw 3-d shapes, calculate area, and calculate volume.

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I can calculate the volume of prisms and cylinder solids.

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I can calculate the area and perimeter of 3-d prisms and cylinders and use the area to find missing length of sides or heights.

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Circumference and Area of Circles - Notes and Practice Problems

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Circumference and Area of Circles - Worksheet and Key

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The circle is arguably the most fundamental shape in our universe, whether you look at the shapes of orbits of planets, whether you look at wheels, whether you look at things on kind of a molecular level. The circle just keeps showing up over and over and over again. So it's probably worthwhile for us to understand some of the properties of the circle. So the first thing when people kind of discovered the circle, and you just have a look at the moon to see a circle, but the first time they said well, what are the properties of any circle? So the first one they might want to say is well, a circle is all of the points that are equal distant from the center of the circle. All of these points along the edge are equal distant from that center right there. So one of the first things someone might want to ask is what is that distance, that equal distance that everything is from the center? Right there. We call that the radius of the circle. It's just the distance from the center out to the edge. If that radius is 3 centimeters, then this radius is going to be 3 centimeters. And this radius is going to be 3 centimeters. It's never going to change. By definition, a circle is all of the points that are equal distant from the center point. And that distance is the radius. Now the next most interesting thing about that, people might say well, how fat is the circle? How wide is it along its widest point? Or if you just want to cut it along its widest point, what is that distance right there? And it doesn't have to be just right there, I could have just as easily cut it along its widest point right there. I just wouldn't be cutting it like some place like that because that wouldn't be along its widest point. There's multiple places where I could cut it along its widest point. Well, we just saw the radius and we see that widest point goes through the center and just keeps going. So it's essentially two radii. You got one radius there and then you have another radius over there. We call this distance along the widest point of the circle, the diameter. So that is the diameter of the circle. It has a very easy relationship with the radius. The diameter is equal to two times the radius. Now, the next most interesting thing that you might be wondering about a circle is how far is it around the circle? So if you were to get your tape measure out and you were to measure around the circle like that, what's that distance? We call that word the circumference of the circle. Now, we know how the diameter and the radius relates, but how does the circumference relate to, say, the diameter. And if you're not really used to the diameter, it's very easy to figure out how it relates to the radius. Well, many thousands of years ago, people took their tape measures out and they keep measuring circumferences and radiuses. And let's say when their tape measures weren't so good, let's say they measured the circumference of the circle and they would get well, it looks like it's about 3. And then they measure the radius of the circle right here or the diameter of that circle, and they'd say oh, the diameter looks like it's about 1. So they would say -- let me write this down. So we're worried about the ratio -- let me write it like this. The ratio of the circumference to the diameter. So let's say that somebody had some circle over here -- let's say they had this circle, and the first time with not that good of a tape measure, they measured around the circle and they said hey, it's roughly equal to 3 meters when I go around it. And when I measure the diameter of the circle, it's roughly equal to 1. OK, that's interesting. Maybe the ratio of the circumference of the diameter's 3. So maybe the circumference is always three times the diameter. Well that was just for this circle, but let's say they measured some other circle here. It's like this -- I drew it smaller. Let's say that on this circle they measured around it and they found out that the circumference is 6 centimeters, roughly -- we have a bad tape measure right then. Then they find out that the diameter is roughly 2 centimeters. And once again, the ratio of the circumference of the diameter was roughly 3. OK, this is a neat property of circles. Maybe the ratio of the circumference to the diameters always fixed for any circle. So they said let me study this further. So they got better tape measures. When they got better tape measures, they measured hey, my diameter's definitely 1. They say my diameter's definitely 1, but when I measure my circumference a little bit, I realize it's closer to 3.1. And the same thing with this over here. They notice that this ratio is closer to 3.1. Then they kept measuring it better and better and better, and then they realized that they were getting this number, they just kept measuring it better and better and they were getting this number 3.14159. And they just kept adding digits and it would never repeat. It was a strange fascinating metaphysical number that kept showing up. So since this number was so fundamental to our universe, because the circle is so fundamental to our universe, and it just showed up for every circle. The ratio of the circumference of the diameter was this kind of magical number, they gave it a name. They called it pi, or you could just give it the Latin or the Greek letter pi -- just like that. That represents this number which is arguably the most fascinating number in our universe. It first shows up as the ratio of the circumference to the diameter, but you're going to learn as you go through your mathematical journey, that it shows up everywhere. It's one of these fundamental things about the universe that just makes you think that there's some order to it. But anyway, how can we use this in I guess our basic mathematics? So we know, or I'm telling you, that the ratio of the circumference to the diameter -- when I say the ratio, literally I'm just saying if you divide the circumference by the diameter, you're going to get pi. Pi is just this number. I could write 3.14159 and just keep going on and on and on, but that would be a waste of space and it would just be hard to deal with, so people just write this Greek letter pi there. So, how can we relate this? We can multiply both sides of this by the diameter and we could say that the circumference is equal to pi times the diameter. Or since the diameter is equal to 2 times the radius, we could say that the circumference is equal to pi times 2 times the radius. Or the form that you're most likely to see it, it's equal to 2 pi r. So let's see if we can apply that to some problems. So let's say I have a circle just like that, and I were to tell you it has a radius -- it's radius right there is 3. So, 3 -- let me write this down -- so the radius is equal to 3. Maybe it's 3 meters -- put some units in there. What is the circumference of the circle? The circumference is equal to 2 times pi times the radius. So it's going to be equal to 2 times pi times the radius, times 3 meters, which is equal to 6 meters times pi or 6 pi meters. 6 pi meters. Now I could multiply this out. Remember pi is just a number. Pi is 3.14159 going on and on and on. So if I multiply 6 times that, maybe I'll get 18 point something something something. If you have your calculator you might want to do it, but for simplicity people just tend to leave our numbers in terms of pi. Now I don't know what this is if you multiply 6 times 3.14159, I don't know if you get something close to 19 or 18, maybe it's approximately 18 point something something something. I don't have my calculator in front of me. But instead of writing that number, you just write 6 pi there. Actually, I think it wouldn't quite cross the threshold to 19 yet. Now, let's ask another question. What is the diameter of the circle? Well if this radius is 3, the diameter is just twice that. So it's just going to be 3 times 2 or 3 plus 3, which is equal to 6 meters. So the circumference is 6 pi meters, the diameter is 6 meters, the radius is 3 meters. Now let's go the other way. Let's say I have another circle. Let's say I have another circle here. And I were to tell you that its circumference is equal to 10 meters -- that's the circumference of the circle. If you were to put a tape measure to go around it and someone were to ask you what is the diameter of the circle? Well, we know that the diameter times pi, we know that pi times the diameter is equal to the circumference; is equal to 10 meters. So to solve for this we would just divide both sides of this equation by pi. The diameter would equal 10 meters over pi or 10 over pi meters. And that is just a number. If you have your calculator, you could actually divide 10 divided by 3.14159, you're going to get 3 point something something something meters. I can't do it in my head. But this is just a number. But for simplicity we often just leave it that way. Now what is the radius? Well, the radius is equal to 1/2 the diameter. So this whole distance right here is 10 over pi meters. If we just 1/2 of that, if we just want the radius, we just multiply it times 1/2. So you have 1/2 times 10 over pi, which is equal to 1/2 times 10, or you just divide the numerator and the denominator by 2. You get 5 there, so you get 5 over pi. So the radius over here is 5 over pi. Nothing super fancy about this. I think the thing that confuses people the most is to just realize that pi is a number. Pi is just 3.14159 and it just keeps going on and on and on. There's actually thousands of books written about pi, so it's not like -- I don't know if there's thousands, I'm exaggerating, but you could write books about this number. But it's just a number. It's a very special number, and if you wanted to write it in a way that you're used to writing numbers, you could literally just multiply this out. But most the time people just realize they like leaving things in terms of pi. Anyway, I'll leave you there. In the next video we'll figure out the area of a circle.

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Radius, diameter, circumference & π (video) | Khan Academy

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The circle is arguably the most fundamental shape in our universe, whether you look at the shapes of orbits of planets, whether you look at wheels, whether you look at things on kind of a molecular level. The circle just keeps showing up over and over and over again.

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Measurements in Similar Figures - MathBitsNotebook(Geo - CCSS Math)

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MathBitsNotebook Geometry CCSS Lessons and Practice is a free site for students (and teachers) studying high school level geometry under the Common Core State Standards.

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Similar Triangles

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Triangle similarity, ratios of area - Math Open Reference

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Similar triangles - ratio of areas is the square of the ratio of the sides.

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Area of Regular Polygons

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Area of Regular Polygons

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Area of Regular Polygons - Worksheet and Key

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Regular polygon area formula - Math Open Reference

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Formula for the area of a regular polygon

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Example

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Exterior Angles of Polygons

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Sal demonstrates how the the sum of the exterior angles of a convex polygon is 360 degrees.

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Interior Angles of Polygons

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Exterior Angles of Polygons

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Polygons Degrees (Angle) Angles Around a Point Interior Angles of Polygons Interior Angles Exterior Angles 2D Shapes Geometry Index

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Sum of Interior Angles of Polygons; Interior Angle Measurement of Regular Polygons

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An Interior Angle is an angle inside a shape

Another example:

Triangles

The Interior Angles of a Triangle add up to 180°

Let's try a triangle:

90° + 60° + 30° = 180°

It works for this triangle

Now tilt a line by 10°:

80° + 70° + 30° = 180°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

Let's try a square:

90° + 90° + 90° + 90° = 360°

Now tilt a line by 10°:

80° + 100° + 90° + 90° = 360°

The Interior Angles of a Quadrilateral add up to 360°

Because there are 2 triangles in a square ...

The interior angles in a triangle add up to 180° ...

... and for the square they add up to 360° ...

... because the square can be made from two triangles!

Pentagon

A pentagon has 5 sides, and can be made from three triangles, so you know what ...

... its interior angles add up to 3 × 180° = 540°

And when it is regular (all angles the same), then each angle is 540 ° / 5 = 108 °
(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)
The Interior Angles of a Pentagon add up to 540°

The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

If it is a Regular Polygon (all sides are equal, all angles are equal)

Shape

Sides

Sum of
Interior Angles

Shape

Each Angle

Triangle
3

180 °

60 °

Quadrilateral
4

360 °

90 °

Pentagon
5

540 °

108 °

Hexagon
6

720 °

120 °

Heptagon (or Septagon)

7

900 °

128.57... °

Octagon

8

1080 °

135 °

Nonagon

9

1260 °

140 °

...

...

..

...

...

Any Polygon
n
( n−2) × 180 °

( n−2) × 180 ° / n

So the general rule is:

Sum of Interior Angles = ( n−2) × 180 °

Each Angle (of a Regular Polygon) = ( n−2) × 180 ° / n

Perhaps an example will help:

Example: What about a Regular Decagon (10 sides) ?

And for a Regular Decagon:

Each interior angle = 1440 °/10 = 144°

Note: Interior Angles are sometimes called "Internal Angles"

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An Interior Angle is an angle inside a shape. Example: ...

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Regular vs. Irregular
Convex vs. Concave
Naming Polygons

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A polygon is aplane shape with straight sides.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon
(straight sides)

Not a Polygon
(has a curve)

Not a Polygon
(open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Regular or Irregular

A regular polygon has all angles equal and all sides equal, otherwise it is irregular

Regular

Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.

If any internal angle is greater than 180° then the polygon is concave. ( Think: concave has a "cave" in it)

Convex

Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon
(this one's a Pentagon)

Complex Polygon
(also a Pentagon)

More Examples

Irregular Hexagon

Concave Octagon

Complex Polygon
(a "star polygon",
in this case a pentagram)

Play With Them!

Try Interactive Polygons ... make them regular, concave or complex.

Names of Polygons

If it is a Regular Polygon...

Name

Sides

Shape

Interior Angle

Triangle (or Trigon)
3
60°

Quadrilateral (or Tetragon)
4
90°

Pentagon 5
108°

Hexagon 6
120°

Heptagon (or Septagon)
7
128.571°

Octagon 8
135°

Nonagon (or Enneagon)
9
140°

Decagon
10
144°

Hendecagon (or Undecagon)
11
147.273°

Dodecagon
12
150°

Megagon
1,000,000
~180°

Googolgon
10¹⁰⁰
~180°

n-gon
n
( n−2) × 180 ° / n

You can make names using this method:

BUT, for polygons with 13 or more sides, it is OK (and easier) to write " 13-gon", " 14-gon" ... " 100-gon", etc.

Remembering

Quadrilateral (4 Sides)

A Quad Bike has 4 wheels

Pentagon (5 Sides)

The " Pentagon" in Washington DC has 5 sides

Hexagon (6 Sides)

Honeycomb has H exagons

Septagon (7 Sides)

Think Septagon is a "Seven- agon"

Octagon (8 Sides)

An Octo pus has 8 tentacles

Nonagon (9 Sides)

Think Nonagon is a "Nine- agon"

Decagon (10 Sides)

Think Decagon has 10 sides,
just like our Dec imal system has 10 digits

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Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon (straight sides) Not a Polygon (has a curve) Not a Polygon (open, not closed) Polygon comes from Greek. Poly- means "many" and -gon means "angle".

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Regular vs. Irregular and Convex vs. Concave

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What's a Polygon (Regular vs. Irregular and Convex vs. Concave Polygons) - Nerdstudy

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Regular, Concave, and Convex Polygons

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SparkNotes: Geometry: Polygons: Different Kinds of Polygons

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A summary of Different Kinds of Polygons in 's Geometry: Polygons. Learn exactly what happened in this chapter, scene, or section of Geometry: Polygons and what it means. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans.

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LT 8.5 Area and Circumference of Circles

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LT 8.5 Area and Circumference of Circles

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I can find the area and circumference of circles and use this to solve other circle problems.

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I can relate areas of similar figures (r²).

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I can calculate the area and perimeter of regular polygons.

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I can find interior and exterior angles of regular polygons related to their sides.

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I can identify special types of polygons (regular, concave, and convex). I can name polygons by the number of sides.

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Let's say that I've just won some type of contest at a car dealership, and they're going to give me a brand new car. And in deciding which car they give me, they're first going to randomly select the engine type. So the engine will come in two different varieties. It'll either be a four-cylinder or a six-cylinder engine. And they're literally just going to flip a fair coin to decide whether I get a four-cylinder engine or a six-cylinder engine. Then they're going to pick the color. And there's four different colors that the cars come in. So I'll write color in a neutral color. So you could get a red car. That's not red. Let me do that in actual red color, or close to red. You could get a red car, you could get a blue car, you could get a green car, or you could get a white car. And once again, they're going to have the red, blue, green, and white in little slips of paper in a bowl and they're just going to pick one of them out. So all of these are equally likely. So given this, that they're just going to flip a coin to pick the engine, and that all of these, the colors all equally likely, I want to think about the probability of getting a six-cylinder white car. So I encourage you to pause the video and think about it on your own. Well, one way to think about this is, well, what are all the equally likely possible outcomes? And then which of those match six-cylinder white car? Well, first, we could think about the engine decision. We're either going to get a four-cylinder engine. So the first decision is the engine. You could view it that way. You're either going to get a four-cylinder engine, or you're going to get a six-cylinder engine. Now, if you got a four-cylinder engine, you're either going to get red, blue, green, or white. And if you've got a six-cylinder engine, once again, you're either going to get red, blue-- I think you see where this is going. That's not blue. Red, blue, green, or white. So how many possible outcomes are there? Well, you could just count. You could kind of say, the leaves of this tree diagram-- one, two, three, four, five, six, seven, eight possible outcomes. And that makes sense. You have two possible engines times four possible colors. You see that right here-- one group of four, two groups of four. So this outcome right here is a four-cylinder blue car. And this outcome over here is a six-cylinder green car. So there's eight equally possible outcomes. And which outcome matches the one that we, I guess, are hoping for, the white six-cylinder car? Well, that's this one right over here. It's one of eight equally likely events. So we have a 1/8 probability. This wasn't the only way that we could have drawn the tree diagram. We could have thought about color as the first row of this tree. So we could have said, look, we're either going to get a-- let me do it down here, so I have a little more space-- we're either going to get a red, a blue-- that's not blue. Changing colors is the hard part. A blue, a green, or a white car. And then for each of those colors, I'm either going to get a four-cylinder or a six-cylinder engine. So it's either going to be four or six. This would be another way of drawing a tree diagram to represent all of the outcomes. So what is this outcome right over here? This is a six-cylinder red car. This is a four-cylinder blue car right over here. Which is the one that we care about? White six-cylinder car? That's this outcome right over here. Once again, you see you have eight equally likely outcomes. And that happens because you have four possible colors. And for each of those four possible colors, you have two different engine types.

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Count outcomes using tree diagram (video) | Khan Academy

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Let's say that I've just won some type of contest at a car dealership, and they're going to give me a brand new car. And in deciding which car they give me, they're first going to randomly select the engine type. So the engine will come in two different varieties.

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Notes and Practice Problems

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Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ... tree diagrams to the rescue!

Here is a tree diagram for the toss of a coin:

The probability of each branch is written on the branch

The outcome is written at the end of the branch

We can extend the tree diagram to two tosses of a coin:

How do we calculate the overall probabilities?

Now we can see such things as:

The probability of "Head, Head" is 0.5×0.5 = 0.25

All probabilities add to 1.0 (which is always a good check)

The probability of getting at least one Head from two tosses is 0.25+0.25+0.25 = 0.75

... and more

That was a simple example using independent events (each toss of a coin is independent of the previous toss), but tree diagrams are really wonderful for figuring out dependent events (where an event depends on what happens in the previous event) like this example:

Example: Soccer Game

You are off to soccer, and love being the Goalkeeper, but that depends who is the Coach today:

with Coach Sam the probability of being Goalkeeper is 0.5

with Coach Alex the probability of being Goalkeeper is 0.3

Sam is Coach more often ... about 6 out of every 10 games (a probability of 0.6).

So, what is the probability you will be a Goalkeeper today?

Let's build the tree diagram. First we show the two possible coaches: Sam or Alex:

The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)

Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):

If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):

The tree diagram is complete, now let's calculate the overall probabilities. This is done by multiplying each probability along the "branches" of the tree.

Here is how to do it for the "Sam, Yes" branch:

(When we take the 0.6 chance of Sam being coach and include the 0.5 chance that Sam will let you be Goalkeeper we end up with an 0.3 chance.)

But we are not done yet! We haven't included Alex as Coach:

An 0.4 chance of Alex as Coach, followed by an 0.3 chance gives 0.12.

Now we add the column:

0.3 + 0.12 = 0.42 probability of being a Goalkeeper today

(That is a 42% chance)

Check

One final step: complete the calculations and make sure they add to 1:

0.3 + 0.3 + 0.12 + 0.28 = 1

Yes, it all adds up.

You can see more uses of tree diagrams on Conditional Probability.

Conclusion

So there you go, when in doubt draw a tree diagram, multiply along the branches and add the columns. Make sure all probabilities add to 1 and you are good to go.

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Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ... tree diagrams to the rescue!

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- [Voiceover] Let's say that I've got a triangle, and this side has length b, which is equal to 12, 12 units or whatever units of measurement we're using. Let's say that this side right over here, this side right over here, has length c, and that happens to be equal to nine. And that we want to figure out the length of this side, and this side has length a, so we need to figure out what a is going to be equal to. Now, we won't be able to figure this out unless we also know the angle here, because you could bring the blue side and the green side close together, and then a would be small, but if this angle was larger than a would be larger. So we need to know what this angle is as well. So let's say that we know that this angle, which we will call theta, is equal to 87 degrees. So how can we figure out a? I encourage you to pause this and try this on your own. Well, lucky for us, we have the Law of Cosines, which gives us a way for determining a third side if we know two of the sides and the angle between them. The Law of Cosines tells us that a squared is going to be equal b squared plus c squared. Now, if we were dealing with a pure right triangle, if this was 90 degrees, then a would be the hypotenuse, and we would be done, this would be the Pythagorean Theorem. But the Law of Cosines gives us an adjustment to the Pythagorean Theorem, so that we can do this for any arbitrary angle. So Law of Cosines tell us a squared is going to be b squared plus c squared, minus two times bc, times the cosine of theta. And this theta is the angle that opens up to the side that we care about. So we can use theta because we're looking for a. If they gave us another angle right over here, that's not the angle that we would use. We care about the angle that opens up into the side that we are going to solve for. So now let's solve for a, because we know what bc and theta actually are. So a squared is going to be equal to b squared... so it's going to be equal to 144, plus c squared which is 81, so plus 81, minus two times b times c. So, it's minus two, I'll just write it out. Minus two times 12 times nine, times the cosine of 87 degrees. And this is going to be equal to, let's see, this is 225 minus, let's see, 12 times nine is 108. 108 times two is 216. Minus 216 times the cosine of 87 degrees. Now, let's get our calculator out in order to approximate this. And remember, this is a squared. Actually, before I get my calculator out, let's just solve for a. So a is just going to be the square root of this. So a is going to be equal to the square root... of all of this business, which I can just copy and paste. It's going to be equal to the square root of that. So let me copy and paste it. So a is going to be equal to the square root of that, which we can now use the calculator to figure out. Let me increase this radical a little bit, so that we make sure we're taking the square root of this whole thing. So let me get my calculator out. So I want to find that square root of 220. Actually, before I do that, let me just make sure I'm in degree mode, and I am in degree mode. Because we're evaluating a trig function in degrees here. So that's fine, so let me exit. So it's going to be 225 minus 216, times cosine of 87 degrees. Not 88 degrees, 87 degrees. And we deserve a drumroll now. This is going to be 14.61, or 14.618. If, say, we wanted to round to the nearest tenth, just to get an approximation, it would be approximately 14.6. So a is approximately equal to 14.6, whatever units we're using long.

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Law of cosines: solving for a side | Trigonometry (video) | Khan Academy

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Voiceover] Let's say that I've got a triangle, and this side has length b, which is equal to 12, 12 units or whatever units of measurement we're using. Let's say that this side right over here, this side right over here, has length c, and that happens to be equal to nine.

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For any triangle:

The Law of Cosines (also called the Cosine Rule) says:

It helps us solve some triangles. Let's see how to use it.

Example: How long is side "c" ... ?

We know angle C = 37º, and sides a = 8 and b = 11

Answer: c = 6.67

How to Remember

How can you remember the formula?

Well, it helps to know it's the Pythagoras Theorem with something extra so it works for all triangles:

Pythagoras Theorem:
(only for Right-Angled Triangles)

So, to remember it:

When to Use

The Law of Cosines is useful for finding:

the third side of a triangle when we know two sides and the angle between them (like the example above)

the angles of a triangle when we know all three sides (as in the following example)

Example: What is Angle "C" ...?

The side of length "8" is opposite angle C, so it is side c. The other two sides are a and b.

Now let us put what we know into The Law of Cosines:

Now we use our algebra skills to rearrange and solve:

In Other Forms

Easier Version For Angles

We just saw how to find an angle when we know three sides. It took quite a few steps, so it is easier to use the "direct" formula (which is just a rearrangement of the formula). It can be in either of these forms:

Example: Find Angle "C" Using The Law of Cosines (angle version)

In this triangle we know the three sides:

Use The Law of Cosines (angle version) to find angle C :

= 57.9° to one decimal place

Versions for a, b and c

Here are all three:

But it is easier to remember the " c²=" form and change the letters as needed !

As in this example:

Example: Find the distance "z"

The letters are different! But that doesn't matter. We can easily substitute x for a, y for b and z for c

Answer: z = 14.5

Did you notice that cos(131º) is negative and this changes the last sign in the calculation to + (plus)? The cosine of an obtuse angle is always negative (see Unit Circle).

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For any triangle: The Law of Cosines (also called the Cosine Rule) says: It helps us solve some triangles. Let's see how to use it. We know angle C = 37º, and sides a = 8 and b = 11 Answer: c = 6.67 How can you remember the formula?

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Voiceover:We've got a triangle here where we know two of the angles and one of the sides. And what I claim, is that I can figure out everything else about this triangle just with this information. You give me two angles and a side, and I can figure out what the other two sides are going to be. And I can, of course, figure out the third angle. So, let's try to figure that out. And the way that we're going to do it, we're going to use something called the Law of Sines. In a future video, I will prove the Law of Sines. But here, I am just going to show you how we can actually apply it. And it's a fairly straightforward idea. The Law of Sines just tells us that the ratio between the sine of an angle, and the side opposite to it, is going to be constant for any of the angles in a triangle. So for example, for this triangle right over here. This is a 30 degree angle, This is a 45 degree angle. They have to add up to 180. So this right over here has to be a, let's see, it's going to be 180 minus 45 minus 30. That's 180 minus 75, so this is going to equal 105 degree angle, right over here. And so applying the Law of Sines, actually let me label the different sides. Let's call this side right over here, side A or has length A. And let's call this side, right over here, has length B. So the Law of Sines tells us that the ratio between the sine of an angle, and that the opposite side is going to be constant through this triangle. So it tells us that sine of this angle, sine of 30 degrees over the length of the side opposite, is going to be equal to sine of a 105 degrees, over the length of the side opposite to it. Which is going to be equal to sine of 45 degrees. equal to the length of the side opposite. So sine of 45 degrees over B. And so if we wanted to figure out A, we could solve this equation right over here. And if we wanted to solve for B, we could just set this equal to that right over there. So let's solve each of these. So what is the sine of 30 degrees? Well, you might just remember it from your unit circles or from even 30, 60, 90 triangles and that's 1/2. And if you don't remember it, you can use a calculator to verify that. I have already verified that this is in degree mode, so it's 0.5. So this is going to be equal to 1/2 over two. So another way of thinking about it, that's going to be equal to 1/4, this piece is equal to 1/4 is equal to sine of a 105 degrees over A. Let me write this, this is equal to sine of 105 degrees over A. And actually, we could also say, since we could actually do both at the same time, that this is equal to that. That 1/4 is equal to sine of 45 degrees over B. Actually, sine of 45 degrees is another one of those that is easy to jump out of unit circles. You might remember it's square of two over two. Let's just write, that's square root of two over two. And you can use a calculator, but you'll get some decimal value right over there. But either case, in either of these equations, let's solve for A then let's solve for B. So one thing we could do is we could take the reciprocal of both sides of this equation. The reciprocal of 1/4 is four. And the reciprocal of this right-hand side is A over the sine of 105 degrees. And then to solve for A, we could just multiply both sides times the sine of a 105 degrees. So we get four times the sine of 105 degrees is equal to A. Let's get our calculator out, so four times the sine of 105 gives us, it's approximately equal to, let's just round to the nearest 100th, 3.86. So A is approximately equal to 3.86. Which looks about right if this is two, and I have made my angles appropriately, that looks like about 3.86. Let's figure out what B is. We could once again take the reciprocal of both sides of this and we get four is equal to B over square root of two over two, we could multiply both sides times square root of two over two. And we would get B is equal to four times the square root of two over two. Come to think of it, B is four times the sine of 45 degrees. Let's figure out what that is. If we wanted actual numerical value, we could just write this as two square roots of two. But let's actually figure out what that is. Two square roots of two is equal to 2.83. So B is approximately equal to 2.83. So [I'm] be clear, this four divided by two is two square roots of two, which is 2.8. Which is approximately equal to 2.83 if we round to the nearest 100th, 2.83, which also seems pretty reasonable here. So the key of the Law of Cosines is if you have two angles and a side, you're able to figure out everything else about it. Or if you actually had two sides and an angle, you also would be able to figure out everything else about the triangle.

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Law of sines: solving for a side | Trigonometry (video) | Khan Academy

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Voiceover:We've got a triangle here where we know two of the angles and one of the sides. And what I claim, is that I can figure out everything else about this triangle just with this information. You give me two angles and a side, and I can figure out what the other two sides are going to be.

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The Law of Sines (or Sine Rule) is very useful for solving triangles:

It works for any triangle:

a, b and c are sides.

A, B and C are angles.
(Side a faces angle A,
side b faces angle B and
side c faces angle C).

And it says that:

When we divide side a by the sine of angle A
it is equal to side b divided by the sine of angle B,
and also equal to side c divided by the sine of angle C

Sure ... ?

Well, let's do the calculations for a triangle I prepared earlier:

a 8 8 0.885... = sin(62.2°) = sin A = 9.04...

b 5 5 0.552... = sin(33.5°) = sin B = 9.06...

c 9 9 0.995... = sin(84.3°) = sin C = 9.04...

The answers are almost the same!
(They would be exactly the same if we used perfect accuracy).

So now you can see that:

Is This Magic?

Not really, look at this general triangle and imagine it is two right-angled triangles sharing the side h:

The sine of an angle is the opposite divided by the hypotenuse, so:

sin(A) = h/b

b sin(A) = h

sin(B) = h/a

a sin(B) = h

a sin(B) and b sin(A) both equal h, so we get:

a sin(B) = b sin(A)

Which can be rearranged to:

We can follow similar steps to include c/sin(C)

How Do We Use It?

Let us see an example:

Example: Calculate side "c"

Now we use our algebra skills to rearrange and solve:

Swap sides: c/sin(105°) = 7/sin(35°)

Calculate: c = ( 7 / 0.574... ) × 0.966...

Finding an Unknown Angle

In the previous example we found an unknown side ...

... but we can also use the Law of Sines to find an unknown angle.

In this case it is best to turn the fractions upside down ( sin A/a instead of a/sin A, etc):

Example: Calculate angle B

Sometimes There Are Two Answers !

There is one very tricky thing we have to look out for:

Two possible answers.

Imagine we know angle A, and sides a and b.

We can swing side a to left or right and come up with two possible results (a small triangle and a much wider triangle)

Both answers are right!

This only happens in the " Two Sides and an Angle not between" case, and even then not always, but we have to watch out for it.

Just think "could I swing that side the other way to also make a correct answer?"

Example: Calculate angle R

The first thing to notice is that this triangle has different labels: PQR instead of ABC. But that's OK. We just use P,Q and R instead of A, B and C in The Law of Sines.

But wait! There's another angle that also has a sine equal to 0.9215...

The calculator won't tell you this but sin(112.9°) is also equal to 0.9215...

So, how do we discover the value 112.9°?

Easy ... take 67.1° away from 180°, like this:

180° − 67.1° = 112.9°

So there are two possible answers for R: 67.1° and 112.9°:

Both are possible! Each one has the 39° angle, and sides of 41 and 28.

So, always check to see whether the alternative answer makes sense.

... sometimes it will (like above) and there are two solutions

... sometimes it won't (see below) and there is one solution

We looked at this triangle before.

As you can see, you can try swinging the "5.5" line around, but no other solution makes sense.

So this has only one solution.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Special Right Triangles - Worksheet and Key

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Special Right Triangles Formulas. 30 60 90 and 45 45 90 special right triangles Examples, Pictures and interactive applet

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Special Right Triangles (30-60-90 and 45,45,90) triangles explained with formulas, examples and pictures

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Special right triangles intro (part 1) (video) | Khan Academy

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Examples, Practice Problems, and Answers

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Examples, Practice Problems, and Answers

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Trigonometry Application - Examples, Practice Problems, and Answers