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Formula For Volume of a Triangular prism: 1/2 bh x h

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Formula For Volume of a Triangular prism: 1/2 bh x h

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1/2 x 10 x 12 x 14= 840
Answer: 840 mm^3

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Shape 2:  A square with  a length of 4 and a width of 4. what would happen if they were doubled?

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Shape 2:  A square with  a length of 4 and a width of 4. what would happen if they were doubled?

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Scale factor 2
4 x 2= 8
4 x 2= 8
8 x 8 = 64
64/16 = 4
4 times greater

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Shape 1: a  rectangle with a length of 2 and width of 3.  What would happen if both dimensions were tripled?

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Shape 1: a  rectangle with a length of 2 and width of 3.  What would happen if both dimensions were tripled?

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scale factor 3
3x3= 9
2 x 3= 6
9x 6= 54
54/6= 9
9 times greater

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EQ2: What you do when the dimensions of a two dimensional shape are changed is find the two scale factors that the dimensions were changed by and multiply them together and then you get your answer.

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EQ2: What you do when the dimensions of a two dimensional shape are changed is find the two scale factors that the dimensions were changed by and multiply them together and then you get your answer.

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1st Square: 3 x 4= 12
2nd Square:  5 x 4= 20
Whole Shape: 20 + 12
Answer: 32 cm^2

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Outside square: 12 x 12 = 144
Inside square: 4 x 4= 16
shaded region: 144-16= 128
Answer: 128 units^2

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The outside square has a length of 12. The square in the middle has a length of 4. If the area outside the middle square is shaded what is the area of the shaded region?

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EQ1: The difference in finding the area is that in inscribed shapes you subtract your two areas in the shape to get the area of the shaded region. While in compound shapes you add your different areas together to get your final area.

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EQ1: The difference in finding the area is that in inscribed shapes you subtract your two areas in the shape to get the area of the shaded region. While in compound shapes you add your different areas together to get your final area.

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Formula For Volume of a Square Pyramid: 1/3 b^2 x h

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Formula For Volume of a Square Pyramid: 1/3 b^2 x h

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1/3 x 16^2 x 15= 1280
Answer: 1280 units^3

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Formula For Surface Area of a Rectangular Prism: 2( L x W + W x H + L x H)

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Formula For Surface Area of a Rectangular Prism: 2( L x W + W x H + L x H)

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2( 8 x 3 + 3 x 5 + 8 x 5)= 158
Answer: 158 in^2

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Formula For Volume of a Cone: 1/3 πr^2 x h

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Formula For Volume of a Cone: 1/3 πr^2 x h

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1/3 x 3.14 x 3^2 x 8=  75.36
Answer: 75.36 in^3

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Formula For Volume of a Cylinder: πr^2 x h

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Formula For Volume of a Cylinder: πr^2 x h

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3.14 x 4^2 x 10= 502.4
Answer: 502.4 in^3

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Formula For Surface Area of a Cylinder: 2πr^2 + h x π x d

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Formula For Surface Area of a Cylinder: 2πr^2 + h x π x d

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2 x 3.14 x 4^2 + 12 x 3.14 x  12= 552.64
Answer: 552.64 in^2

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Formula For Circumference of a Circle: D x  π

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Formula For Circumference of a Circle: D x  π

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D= r x 2
D=  7
3.14 x 7= 21.98
Answer: 21.98 cm^2

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Formula For Area of a Parallelogram: b x h

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Formula For Area of a Parallelogram: b x h

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20 x 30 = 600
Answer: 600 units^

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3.14  x 3.5^2 = 38.465
Answer: 38.465 cm^2

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2.5 x 4= 10
Answer: 10m^2

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Formula For Surface Area of a Square Pyramid: bh + bh x 2

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Formula For Surface Area of a Square Pyramid: bh + bh x 2

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5 x 4.5 + 5 x 4.5 x 2= 67.5
Answer: 67.5 in^2

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4/3 x 3.14 x 5^3= 523.33333
Answer: 523.3m^3

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Formula For Surface Area of a  Triangular Prism: bh+H(S1+S2+S3)

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Formula For Surface Area of a  Triangular Prism: bh+H(S1+S2+S3)

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16 x 12 + 10(16+20+12)=  672
Answer: 672 cm^2

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Formula For Area of a Trapezoid: 1/2 h(b1+ b2)

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Formula For Area of a Trapezoid: 1/2 h(b1+ b2)

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1/2 x 4(12+6)= 36
Answer: 36m^2

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Jim has a toy house set that he has to put together. If his triangular roof is 3 inches long and the height is 2 what is the area?
1/2 x 3 x 2
Answer: 3 units^2

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- Website for prime coordinates is incorrect.

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When Rotating 90* Counter clockwise your y always changes and your x stays the same. (x,y)--->     (x,-y). When rotating 90* clockwise then your x changes instead of your y changing. (x,y)----> (-x,y). If rotated 180* then x and y changes.

When Reflecting the Y axis then your x axis will change and your y axis will stay the same. (x,y)--->(-x,y). If you reflect the x axis then your x axis stays the same and your y axis changes. (x,y)---->(x,-

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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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Prime Coordinate: The coordinates on a graph after an transformation

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Prime Coordinate: The coordinates on a graph after an transformation

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"Rotation" means turning around a center:

The distance from the center to any point on the shape stays the same.

Every point makes a circle around the center:

Here a triangle is rotated around
the point marked with a "+"

Try It Yourself

Here you can drag the pin and try different shapes:

And here you can choose an angle and see how to rotate different shapes point-by-point. Try and follow what happens each time:

6712, 6713, 6714, 6715, 839, 840, 2133, 2134, 3340, 3341

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The distance from the center to any point on the shape stays the same. Every point makes a circle around the center: Here a triangle is rotated around the point marked with a "+" Here you can drag the pin and try different shapes: And here you can choose an angle and see how to rotate different shapes point-by-point.

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Rotation: A transformation preformed by spinning  the figure around a fixed point. The fixed point is called the center of rotation

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Rotation: A transformation preformed by spinning  the figure around a fixed point. The fixed point is called the center of rotation

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Reflections are everywhere ... in mirrors, glass, and here in a lake.
... what do you notice ?

Every point is the same distance from the central line !
... and ...
The reflection has the same size as the original image.

The central line is called the Mirror Line:

Can A Mirror Line Be Vertical?

Yes. Here my dog "Flame" shows a Vertical Mirror Line (with a bit of photo editing).

You can try reflecting some shapes about different mirror lines here:

How Do I Do It Myself?

Just approach it step-by-step. For each corner of the shape:

1. Measure from the point to the mirror line (must hit the mirror line at a right angle)

2. Measure the same distance again on the other side and place a dot.

3. Then connect the new dots up!

Labels

It is common to label each corner with letters, and to use a little dash (called a Prime) to mark each corner of the reflected image.

Here the original is ABC and the reflected image is A'B'C'

Some Tricks

X-Axis

When the mirror line is the x-axis
we change each (x,y) into(x,−y)

Y-Axis

When the mirror line is the y-axis
we change each (x,y) into(−x,y)

Fold the Paper

And when all else fails, just fold the sheet of paper along the mirror line and then hold it up to the light !

6716, 6717, 3346, 3344, 3345, 3347, 5152, 5153, 841, 842

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Learn about reflection in mathematics: every point is the same distance from a central line.

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Reflection: A flipping of an object over a line which is known as the line of reflection.

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Reflection: A flipping of an object over a line which is known as the line of reflection.

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( tap on image)

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In Geometry, "Translation" simply means Moving ...

... without rotating, resizing or anything else, just moving.

To Translate a shape:

Every point of the shape must move:

the same distance

in the same direction.

To see how this works, try translating different shapes here:

Note: You can translate either by angle-and-distance, or by x-and-y.

Try both to see what happens.

Writing it Down

Sometimes we just want to write down the translation, without showing it on a graph.

Example: to say the shape gets moved 30 Units in the "X" direction, and 40 Units in the "Y" direction, we can write:

(x,y) → (x+30,y+40)

Which says "all the x and y coordinates become x+30 and y+40"

6720, 6721, 6722, 6726, 3348, 3349, 3350, 843, 2135, 2136

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In Geometry, "Translation" simply means Moving ... ... without rotating, resizing or anything else, just moving. To Translate a shape: Every point of the shape must move: the same distance in the same direction. To see how this works, try translating different shapes here: Note: You can translate either by angle-and-distance, or by x-and-y.

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Translation: Moving an object so that every point of the object moves in the same direction as well as the same distance.

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Translation: Moving an object so that every point of the object moves in the same direction as well as the same distance.

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

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Dilation: A transformation that moves each point along a ray which starts from a fixed center, and multiplies distances from the center as a common factor. 

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Dilation: A transformation that moves each point along a ray which starts from a fixed center, and multiplies distances from the center as a common factor. 

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- Corresponding angles definition is missing that they are in the same relative location and are equal.

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In this example, these are corresponding angles:

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Corresponding Angles: When two lines are crossed by another line called a transversal 

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Corresponding Angles: When two lines are crossed by another line called a transversal 

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Angles a° and c° are also vertical angles, so must be equal, which means they are 140° each.

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Two angles are Adjacent when they have a common side and a common vertex (corner point) and don't overlap.

Angle ABC is adjacent to angle CBD

Because:

they have a common side (line CB)

they have a common vertex (point B)

What Is and Isn't an Adjacent Angle

These ARE Adjacent Angles

they share a vertex and a side

NOT Adjacent Angles

they only share a vertex, not a side

NOT Adjacent Angles

they only share a side,
not a vertex

Don't Overlap!

ALSO the angles must not overlap.

NOT Adjacent Angles

angles a and b overlap

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Two angles are Adjacent when they have a common side and a common vertex (corner point) and don't overlap. Angle ABC is adjacent to angle CBD Because: they have a common side (line CB) they have a common vertex (point B) These ARE Adjacent Angles they share a vertex and a side NOT Adjacent Angles they only share a vertex, not a side NOT Adjacent Angles they only share a side, not a vertex ALSO the angles must not overlap.

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These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:

Notice that together they make a straight angle.

But the angles don't have to be together.

These two are supplementary because
60° + 120° = 180°

Play With It ...

(Drag the points)

images/geom-comp-supp.js?mode=supp

When the two angles add to 180°, we say they "Supplement" each other. comes from Latin supplere, to complete or "supply" what is needed

Complementary vs Supplementary

A related idea is Complementary Angles, they add up to 90°

How to remember which is which? Well, alphabetically they are:

Complementary add to 90°

Supplementary add to 180°

You can also think:

"C" of Complementary is for "Corner" (a Right Angle), and

"S" of Supplementary is for "Straight" (180° is a straight line)

Or you can think:

when you are right you get a compliment (sounds like complement)

"supplement" (like a vitamin supplement) is something extra, so is bigger

817, 1804, 5232, 5234, 818, 3303, 1805, 3304, 5235, 5236

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These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°: Notice that together they make a straight angle. But the angles don't have to be together. These two are supplementary because 60° + 120° = 180° (Drag the points) images/geom-comp-supp.js?mode=supp When the two angles add to 180°, we say they "Supplement" each other.

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Two angles are Complementary when they
add up to 90 degrees (a Right Angle )

These two angles (40° and 50°) are Complementary Angles, because they add up to 90°:

Notice that together they make a right angle

But the angles don't have to be together.

These two are complementary because 27° + 63° = 90°

Play With It ...

(Drag the points)

images/geom-comp-supp.js?mode=comp

Right Angled Triangle

In a right angled triangle, the two non-right angles are complementary, because in a triangle the three angles add to 180°, and 90° has already been taken by the right angle.

When two angles add to 90°, we say they "Complement" each other.

Complementary comes from Latin completum meaning "completed" ...
because the right angle is thought of as being a complete angle.

Spelling: be careful, it is not "Complimentary Angle" (with an "i") ...
that would be an angle you get for free!

Complementary vs Supplementary

A related idea is Supplementary Angles - those add up to 180°

How to remember which is which? Well, alphabetically they are:

Complementary add to 90°

Supplementary add to 180°

You can also think:

"C" of Complementary is for "Corner" (a Right Angle), and

"S" of Supplementary is for "Straight" (180° is a straight line)

Or you can think:

when you are right you get a compliment (sounds like complement)

"supplement" (like a vitamin supplement) is something extra, so is bigger

5231, 5233, 819, 2109, 820, 2110, 5240, 9063, 3305, 3306

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Two angles are Complementary when they add up to 90 degrees (a Right Angle )

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Adjacent Angles: Two angles that share the same vertex and one ray

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Adjacent Angles: Two angles that share the same vertex and one ray

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Vertical Angles: Two angles that are opposite from each other when two lines cross. They share the same vertex

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Vertical Angles: Two angles that are opposite from each other when two lines cross. They share the same vertex

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Supplementary Angles: Two angles whose sum is equal to 180 

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Supplementary Angles: Two angles whose sum is equal to 180 

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Complementary Angles: Two angles whose sum is equal to 90

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Complementary Angles: Two angles whose sum is equal to 90

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Triangle Sum Theorem: The sum of interior  angles that add up to 180 degrees.   

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Triangle Sum Theorem: The sum of interior  angles that add up to 180 degrees.   

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Proportional Reasoning and Linear Equations Grade - 86%

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Proportional Reasoning and Linear Equations Grade - 86%

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- Slope values need to be given for 0 and undefined, Not just an example of an equation.  y=x+6 would be an example of a positive slope also, not zero.
- Missing the linear equation y=mx+b and that the m is the slope and the b is the y-intercept.
- Similar figures also must have congruent corresponding angles and the same shape, but different size.

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The parts of a linear equation are the x axis and the y axis. You can find the x axis in the line that goes across the chart. You can find the y axis on the line that goes up the chart

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The parts of a linear equation are the x axis and the y axis. You can find the x axis in the line that goes across the chart. You can find the y axis on the line that goes up the chart

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To see if  two figures are similar one they most be different sizes. Also you have to see if the length and width of each figure can go into each other.

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To see if  two figures are similar one they most be different sizes. Also you have to see if the length and width of each figure can go into each other.

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In this photo the rectangles are different sizes but if multiplied by two then they would be the same

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The Constant Of Proportionality - Proportional Relationships

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In this video, we discuss what the constant of proportionality is, how to find it, as well as its uses. We also demonstrate how it can be used with a table. On EDU Chronicles, we upload videos based on different educational subjects for students and educators.

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Constant of Proportionality-  the constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the factor of proportionality

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Constant of Proportionality-  the constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the factor of proportionality

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Using a Scale Factor to Solve Ratio Problems

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How to solve ratio problems using a scale factor

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Scale Factor-             The ratio of any two corresponding lengths in two similar geometric figures

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Scale Factor-             The ratio of any two corresponding lengths in two similar geometric figures

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A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways:

two equal fractions,

or,

using a colon, a:b = c:d

When two ratios are equal, then the cross products of the ratios are equal.

That is, for the proportion, a:b = c:d , a x d = b x c

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Ratios and Proportions - Proportions - First Glance

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A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways: two equal fractions, or, using a colon, a:b = c:d When two ratios are equal, then the cross products of the ratios are equal.

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Proportion-a part, share, or number considered in comparative relation to a whole

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Proportion-a part, share, or number considered in comparative relation to a whole

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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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Unit Rate- The comparison of an amount to a measurement 

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Unit Rate- The comparison of an amount to a measurement 

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x= -3

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y= x+6

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y= -3/4x

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y= 5/5x+1

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- Missing the question for why you flip an inequality symbol.

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Less than or equal to      No more than             Maximum

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Less than or equal to      No more than             Maximum

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You owe Big Cheese a maximum of $20
<--------.
|----------|-----------|
19 20 21

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Greater than or equal to                                  No less than                  Minimum

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Greater than or equal to                                  No less than                  Minimum

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You must be no less than 60 in. to ride the little kid ponie ride

.----->
|--------|--------|
59 60 61

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Less than                       Below                        Lower than                           

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Less than                       Below                        Lower than                           

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You have less than 85 slices of pizza today for lunch

<------- 0
|---------|---------|
84         85          86

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0--------->
|------------|-----------|
54             55             56

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Greater than                 More than                     Above

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Greater than                 More than                     Above

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You have more than 55 donuts in your mouth

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Beginning Algebra & Solving Algebraic Equations

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Number of Solutions for Systems of Equations

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What is a "Solution Set" for an Equation (Algebra)

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Subscribe on YouTube: http://bit.ly/1bB9ILD Leave some love on RateMyProfessor: http://bit.ly/1dUTHTw Send us a comment/like on Facebook: http://on.fb.me/1eWN4Fn

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Sometimes we need to solve Inequalities like these:

Solving

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

Simplify:

x > 10
Solved!
How to Solve

Solving inequalities is very like solving equations, we do most of the same things ...

... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"

Some things can change the direction!

Safe Things To Do

These things do not affect the direction of the inequality:

Add (or subtract) a number from both sides

Multiply (or divide) both sides by a positive number

Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

3x < 10

But these things do change the direction of the inequality ("<" becomes ">" for example):

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x < 4

And that is our solution: x < 4

In other words, x can be any value less than 4.

What did we do?

We went from this:

To this:

x+3 < 7

x < 4

And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5

If we subtract 5 from both sides, we get:

7 < x

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number:

Example: 3y < 15

If we divide both sides by 3 we get:

y < 5

And that is our solution: y < 5

Negative Values

When we multiply or divide by a negative number
we must reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase,
but from −3 to −7 is a decrease.

−7 < −3

7 > 3

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality!

−2y < −8

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Example: bx < 3b

It seems easy just to divide both sides by b, which gives us:
x < 3
... but wait ... if b is negative we need to reverse the inequality like this:
x > 3
But we don't know if b is positive or negative, so we can't answer this one!

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b:

The answer could be x < 3 or x > 3 and we can't choose because we don't know b.
So:
Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 < −10

Now add 3 to both sides:

x < −7

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Example:

First, let us clear out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

−6 < 6−2x < 12

Now subtract 6 from each part:

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction.

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

Summary

Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.

But these things will change direction of the inequality:

Multiplying or dividing both sides by a negative number

Swapping left and right hand sides

Don't multiply or divide by a variable (unless you know it is always positive or always negative)

447,448,303,304,449,450, 1222, 1223, 1224, 1225

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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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Answer(s) that allow an inequality statement to be true

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Comparing two things using symbols of < > </= >/=

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- When estimating a square root you would think about the perfect square that is above and below the number. Ex. 56 is between 49 and 64 there for it is about 7.5.

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A real number that can NOT be made by dividing two integers (an integer has no fractional part).

"Irrational" means "no ratio", so it isn't a rational number. We aren't saying it's crazy!

Its decimal also goes on forever without repeating.

Example: π (the famous number "pi") is an irrational number, as it can not be made by dividing two integers.

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A real number that can NOT be made by dividing two integers (an integer has no fractional part). "Irrational" means "no ratio", so it isn't a rational number. We aren't saying it's crazy! Its decimal also goes on forever without repeating.

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Irrational Numbers: A real number that can not be made by dividing two integers.

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Irrational Numbers: A real number that can not be made by dividing two integers.

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

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Repeated Decimals:a decimal numeral that, after a certain point, consists of a group of one or more digits repeated.

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Repeated Decimals:a decimal numeral that, after a certain point, consists of a group of one or more digits repeated.

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How to Cube A Number

To cube a number, just use it in a multiplication 3 times ...

Example: What is 3 Cubed?

3 Cubed

=

=

3 × 3 × 3

= 27

Note: we write "3 Cubed" as (the little ³ means the number appears three times in multiplying)

Cube Root

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

3

27

The cube root of a number is ...... a special value that when
cubed gives the original number.

Note: When you see "root" think

"I know the tree, but what is the root that produced it?"

In this case the tree is "27", and the cube root is "3".

Here are some cubes and cube roots:

Example: What is the Cube root of 125?

Well, we just happen to know that 125 = 5 × 5 × 5 (if we use 5 three times in a multiplication we get 125) ...
... so the cube root of 125 is 5
The Cube Root Symbol

This is the special symbol that means "cube root", it is the "radical" symbol (used for square roots) with a little three to mean cube root.

You can use it like this:

(we say "the cube root of 27 equals 3")

You Can Also Cube Negative Numbers

Have a look at this:

So the cube root of −125 is −5

Perfect Cubes

The Perfect Cubes are the cubes of the whole numbers:

It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

Example: what is the cube root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

We are getting closer, but very slowly ... at this point, I get out my calculator and it says:

... but the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation !

(Further reading: these kind of numbers are calledsurds which are a special type of irrational number)

316, 2007, 2008, 5081, 5084, 3157, 3158, 5082, 5083, 5085

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To cube a number, just use it in a multiplication 3 times ... 3 Cubed = = 3 × 3 × 3 = 27 Note: we write "3 Cubed" as (the little 3 means the number appears three times in multiplying) A cube root goes the other direction: 3 cubed is 27, so the cube root of 27 is 3 3 27 The cube root of a number is ......

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Cube Root: the number that produces a given number when cubed

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Cube Root: the number that produces a given number when cubed

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First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number: multiply it by itself.

Example: What is 3 squared?

3 Squared

=

= 3 × 3 = 9

"Squared" is often written as a little 2 like this:

This says"4 Squared equals 16"(the little 2 says the number appears twice in multiplying)

Negative Numbers

We can also square negative numbers.

That was interesting!

When we square a negative number we get a positive result.

Just the same as squaring a positive number:

(For more detail read Squares and Square Roots in Algebra)

Square Roots

A square root goes the other way:

3 squared is 9, so a square root of 9 is 3

A square root of a number is ...

... a value that can be multiplied by itself to give the original number.

A square root of 9 is ...

It is like asking:

What can we multiply by itself to get this?

To help you remember think of the root of a tree:

"I know the tree, but what root made it?"

In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

4

16

5

25

36

49

Decimal Numbers

It also works for decimal numbers.

Try the sliders below (note: '...' means the decimals continue on forever):

Using the sliders:

Negatives

We discovered earlier that we can square negative numbers:

Example: (−3) squared

(−3) × (−3) = 9

And of course 3 × 3 = 9 also.

So the square root of 9 could be −3 or +3

Example: What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

So the square roots of 25 are −5 and +5

The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick,
and actually started hundreds of years ago as a dot with a flick upwards.

It is called theradical, and always makes mathematics look important!

We use it like this:

and we say"square root of 9 equals 3"

Example: What is √25?

25 = 5 × 5, in other words when we multiply 5 by itself (5 × 5) we get 25
So the answer is:
√25 = 5

But wait a minute! Can't the square root also be −5? Because (−5) × (−5) = 25 too.

Well the square root of 25 could be −5 or +5.

But when we use the radical symbol √ we only give the positive (or zero) result.

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The Perfect Squares (also called "Square Numbers") are the squares of the integers:

Try to remember them up to 12.

Calculating Square Roots

It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

Example: what is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:

3.1622776601683793319988935444327

But the digits just go on and on, without any pattern.

So even the calculator's answer is only an approximation !

Note: numbers like that are called Irrational Numbers, if you want to know more.
The Easiest Way to Calculate a Square Root

Use your calculator's square root button!

And also use your common sense to make sure you have the right answer.

A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

a) start with a guess (let's guess 4 is the square root of 10)

b) divide by the guess (10/4 = 2.5)
c) add that to the guess (4 + 2.5 = 6.5)
d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)
e) now, set that as the new guess , and start at b) again

And so, after 3 times around the answer is 3.1623, which is pretty good, because:
3.1623 x 3.1623 = 10.00014
Now ... why don't you try calculating the square root of 2 this way?

How to Guess

What if we have to guess the square root for a difficult number such as "82,163" ... ?

In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 is a good start.

Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like 4/4/16

The next after that is the 5th of May 2025 (5/5/25)

309,310,315, 1082, 1083, 2040, 3156, 2041, 2042, 3154

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First learn about Squares, then Square Roots are easy. To square a number: multiply it by itself. 3 Squared = = 3 × 3 = 9 "Squared" is often written as a little 2 like this: This says"4 Squared equals 16" (the little 2 says the number appears twice in multiplying) We can also square negative numbers.

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Square Root: a number that produces a specified quantity when multiplied by itself

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Square Root: a number that produces a specified quantity when multiplied by itself

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EQ1:To convert a repeating decimal to a fraction you must fist start by turning your repeating digits into x. Then Put the repeating digits in front of were the digits start and when the first digits end.  You may have to move your decimal point to do this so each decimal point you move is times 10x. After that subtract your two x's from each other and then divide the remaining x's by the amount of remaining x's and also divide the amount of repeating digits by the amount of x's and you will get your answer.   EXAMPLE:

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EQ1:To convert a repeating decimal to a fraction you must fist start by turning your repeating digits into x. Then Put the repeating digits in front of were the digits start and when the first digits end.  You may have to move your decimal point to do this so each decimal point you move is times 10x. After that subtract your two x's from each other and then divide the remaining x's by the amount of remaining x's and also divide the amount of repeating digits by the amount of x's and you will get your answer.   EXAMPLE:

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Converting Repeating Decimals to Fractions

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EQ2: To estimate the square root of a number you must think of what m=number multiplied by itself might equal the number or close to the number since it's just an estimate. For example if we were to estimate the square root of 81 then we would start thinking about what number multiplied by itself is 81 or close to 81. After thinking you would realize that the number that goes in when multiplied by itself is 9 because 9 x 9 = 81

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EQ2: To estimate the square root of a number you must think of what m=number multiplied by itself might equal the number or close to the number since it's just an estimate. For example if we were to estimate the square root of 81 then we would start thinking about what number multiplied by itself is 81 or close to 81. After thinking you would realize that the number that goes in when multiplied by itself is 9 because 9 x 9 = 81