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What's the Difference Between Square Footage and Cubic Feet? – Construction Math

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Easiest way to Learn Volume of Cylinder, Cone, Sphere and Hemisphere

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Easiest way to Learn Volume of Cylinder, Cone, Sphere and Hemisphere

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Easiest way to Learn Volume of Cylinder, Cone, Sphere and Hemisphere

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one way : cones and spheres
another way: Pyramids and spheres

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But why is a sphere's surface area four times its shadow?

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Cavalieri's Principle in 3D | Volume of a sphere |

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How to triple your memory by using this trick | Ricardo Lieuw On | TEDxHaarlem

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Debateable 2D cross section of 3D objects video

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Debateable 2D cross section of 3D objects video

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What does this video do a good job of explaining? What does this video do a bad job? Is anything misleading or inaccurate?

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3D Shapes into 2D Objects | SliceMaster 2000 | PBSMathClub

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2D cross section of 3D objects- application

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2D cross section of 3D objects- application

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CT scan image.... @ 1:14

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2D cross section of 3D objects application

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2D cross section of 3D objects application

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math of CT Scan

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Basic mathematics of Computed Tomography

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Generate the sine graph from the unit circle. See also: [b]Unit Circle and Cosine Graph: [/b]http://www.geogebratube.org/material/show/id/8029 [b...

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Visual Exploration...No words

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Pythagorean Theorem Visual Proof- No Words

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Pythagorean Theorem Visual Proof- No Words

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Drag the points in the sketch below.
What do you notice? What does this prove? How does this prove it? This is purposefully vague to let kids explor…

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PEOPLE DENSITY Per Country | Population Comparision

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What is Density? | Gravitation | Physics | Infinity Learn

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Pythaogrean Theorem is the Distance Formula

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Pythaogrean Theorem is the Distance Formula

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- We are asked what is the distance between the following points. Pause this video and see if you can figure it out. There's multiple ways to think about it. The way I think about it is really to try to draw a right triangle where these points, where the line that connects these points is the hypotenuse and then we can just use the Pythagorean Theorem. Let me show you what I am talking about. Let me draw a right triangle, here. That is the height of my right triangle and this is the width of my right triangle. Then the hypotenuse will connect these two points. I could use my little ruler tool here to connect that point and that point right over there. I'll color it in orange. There you have it. There you have it. I have a right triangle where the line that connects those two points is the hypotenuse of that right triangle. Why is that useful? From this, can you pause the video and figure out the length of that orange line, which is the distance between those two points? What is the length of this red line? You could see it on this grid, here. This is equal to two. It's exactly two spaces, and you could even think about it in terms of coordinates. The coordinate of this point up here is negative five comma eight. Negative five comma eight. The coordinate here is X is four, Y is six. Four comma six, and so the coordinate over here is going to have the same Y coordinate as this point. This is going to be comma six. It's going to have the same X coordinate as this point. This is going to be negative five comma six. Notice, you're only changing in the Y direction and you're changing by two. What's the length of this line? You could count it out, one, two, three, four, five, six, seven, eight, nine. It's nine, or you could even say hey look, we're only changing in the X value. We're going from negative five, X equals negative five, to X equals four. We're going to increase by nine. All of that just sets us up so that we can use the Pythagorean Theorem. If we call this C, we know that A squared plus B squared is equal to C squared, or we could say that two squared ... Let me do it over here. Use that same red color. Two squared plus nine squared, plus nine squared, is going to be equal to our hypotenuse square, which I'm just calling C, is going to be equal to C squared, which is really the distance. That's what we're trying to figure out. Two squared, that is four, plus nine squared is 81. That's going to be equal to C squared. We get C squared is equal to 85. C squared is equal to 85 or C is equal to the principal root of 85. Can I simplify that a little bit? Let's see. How many times does five go into 85? It goes, let's see, it goes 17 times. Neither of those are perfect squares. Yeah, that's 50 plus 35. Yeah, I think that's about as simple as I can write it. If you wanted to express it as a decimal, you could approximate it by putting this into a calculator and however precise you want your approximation to be. That over here, that's the length of this line, our hypotenuse and our right triangle, but more importantly for the question they're asking, the distance between those points.

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Finding distance with Pythagorean theorem (video) | Khan Academy

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We are asked what is the distance between the following points. Pause this video and see if you can figure it out. There's multiple ways to think about it. The way I think about it is really to try to draw a right triangle where these points, where the line that connects these points is the hypotenuse and then we can just use the Pythagorean Theorem.

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How to find the perimeter of a trianlge using the distance formula

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How to find the perimeter of a trianlge using the distance formula

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How to Find the Perimeter Using the Distance Formula : Math Calculations

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Maximizing Area of 4-Sided Rectangular Enclosures With a Fixed Perimeter | Optimization

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Transversals and their properties:
https://www.youtube.com/watch?v=6RMN5Pf1fHU

All angles in a triangle adding up to 180
https://www.youtube.com/watch?v=_PnPM8VVHBA&t=11s

Pythagorean Theorem
Part 1: https://www.youtube.com/watch?v=RGh8n86EE48
Part 2:
https://www.youtube.com/watch?v=KnEgt5V2H1U

interior and exterior angles in a triangle:
https://www.youtube.com/watch?v=EZ6dOlRQDBo

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Geometry 9.5: Properties of diagonals of quadrilaterals

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Properties of Parallelograms (with practice)

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What is a Parallelogram? | Special Cases of Parallelogram | Don't Memorise

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All quadrilaterals are coplanar and have two diagonals.

Quadrilaterals may be drawn in a "convex" form or a "concave" form. The quadrilaterals we will be using will be convex, where the diagonals lie within the figure. A concave quadrilateral contains an angle greater than 180º, and at least one of the diagonals lies partially, or entirely, outside of the figure (as shown below). The concave figure "caves in" upon itself.

There are several "types" of quadrilaterals determined by their characteristics, or properties.
We will be examining three types of quadrilaterals: parallelograms, trapezoids and kites.

There are three additional quadrilaterals that possess the properties of a parallelogram.

Definition: A parallelogram has two pairs of parallel sides.
* The opposite sides of a parallelogram are congruent.
* A parallelogram has no reflectional symmetry.
* A parallelogram has rotational symmetry of 180º (Order 2).

Definition: A rectangle is a parallelogram with 4 right angles.
* A rectangle has two pairs of parallel sides.
* A rectangle has opposite sides congruent.
* A rectangle also has four right angles.
* A rectangle has two axes of reflectional symmetry.
* A rectangle has rotational symmetry of 180º (Order 2).

Definition: A rhombus is a parallelogram with 4

sides.2 (diagonals) axes of reflectional symmetry.
* A rhombus has two pairs of parallel sides.
* A rhombus has all four sides congruent.
* A rhombus has
* A rhombus has rotational symmetry of 180º (Order 2).

Definition: A square is a parallelogram with 4

sides and 4 right angles.
* A square has two pairs of parallel sides.
* A square has all four sides congruent.
* A square has four right angles.
* A square has four axes of reflectional symmetry.
* A square has rotational symmetry of 90º (Order 4).

There are two basic types of trapezoids: regular trapezoids and isosceles trapezoids.

Definition: A trapezoid has at least 1 pair of parallel sides.
* A trapezoid has no axes of reflectional symmetry.
* A trapezoid has no rotational symmetry (Order 1).
Note: Having no rotational symmetry means that the figure must be rotated a full 360º to again appear in it original position. This may be referred to as the identity symmetry.

Definition: An isosceles trapezoid is a trapezoid with congruent base angles.
* An isosceles trapezoid has congruent legs.
* An isosceles trapezoid has at least 1 axis of reflectional symmetry. (If it has 2, it is a rectangle.)
* An isosceles trapezoid (with only 1 pair of parallel sides) has no rotational symmetry.

Note: The definition of an isosceles triangle states that the triangle has two congruent "sides". But the definition of isosceles trapezoid stated above, mentions congruent base "angles", not sides (or legs). Why?
If an isosceles trapezoid is defined to be "a trapezoid with congruent legs", a parallelogram will be an isosceles trapezoid. If this occurs, the other properties that an isosceles trapezoid can possess can no longer hold, since they will not be true for a parallelogram.
If we define an isosceles trapezoid to be a trapezoid with congruent base angles, we can easily prove the sides (legs) to also be congruent, a parallelogram will NOT be an isosceles trapezoid, and all of the commonly known properties of an isosceles trapezoid will still be true.

* A quadrilateral kite resembles a flying kite.

* A kite has two pairs of adjacent sides of equal measure. Notice that these sides are not "opposite" sides.

* The kite at the right has one vertical axis of reflectional symmetry, .

* The kite has no rotational symmetry.

A kite also resembles a triangle reflected over one of its sides, forming a quadrilateral.
Notice Δ ABD being reflected over its side from B to C, forming Δ CBD.

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General Quadrilateral Information - MathBitsNotebook(Geo - CCSS Math)

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All quadrilaterals are coplanar and have two diagonals. Quadrilaterals may be drawn in a "convex" form or a "concave" form. The quadrilaterals we will be using will be convex, where the diagonals lie within the figure. A concave quadrilateral contains an angle greater than 180º, and at least one of the diagonals lies partially, or entirely, outside of the figure (as shown below).

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6-8: Regular Polygon Rotational Symmetry: Khan Academy Lesson with visuals

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6-8: Regular Polygon Rotational Symmetry: Khan Academy Lesson with visuals

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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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6-8: Regular Polygon Rotational Symmetry: Quick and to the Point visual

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6-8: Regular Polygon Rotational Symmetry: Quick and to the Point visual

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Order of rotational symmetry of regular polygons

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Order of rotational symmetry of regular polygons

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Visuals for 6-7: Rotational and Reflection Symmetry

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Visuals for 6-7: Rotational and Reflection Symmetry

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Rotational and reflection symmetry:

https://www.youtube.com/watch?v=blQjglPb1jw

https://www.youtube.com/watch?v=s4tS-ZmpJfw

https://www.youtube.com/watch?v=xVYa9orJv08

This last one is OK but not great because the shapes are not actually rotated with a program:  https://www.youtube.com/watch?v=273Uw7UgzL4

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5-7 Why does the sum of all exterior angles add up to 360 degrees, regardless of the number of sides a figure is?

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5-7 Why does the sum of all exterior angles add up to 360 degrees, regardless of the number of sides a figure is?

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1) Open Link
2) Pick a "number of sides"
3) Toggle Exterior Angles Icon to the right (to turn it ON)
4) Make the radius go from big to zero...see how the exterior angles connect and make a circle!!!!

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Regular Polygons

To create a regular polygon simply drag one of the three sliders. The sides slider changes the number of sides. For a regular polygon all the vertices lie on a circle circumference. The radius slider adjusts this circle radius. You can display the circle which is initially transparent. Simply click the circle background ctrl followed by a colour from the colour picker. To hide the circle click the X in the colour picker. The angle slider adjusts the overall orientation of polygon.

Irregular Polygons

All the vertices can be clicked and dragged to new positions. You can turn off this feature by clicking the drag toggle button. You can also change the snap value, when set to one the points snaps to the nearest square, set to zero for no snapping.

A quick way to create an irregular polygon is start with a regular one and then click the dice which move all the vertices by a small random amount. Click repeatedly and the polygon wobbles like a jelly. You can add a side/vertex by clicking the + button, a new vertex point will be inserted at the mid point of the side clockwise of the currently selected point. To remove a vertex click the - button.

Measuring Angles or Sides

The activity can display a protractor for measuring angles, click the protractor button to display or hide it. You can also display a ruler by clicking the ruler button. Note when you are using the the protractor or ruler it is best to turn off draggable points.

Displaying interior and exterior angles automatically

Various angle types can be displayed. Start by clicking the interior toggle. The interior angle appears, to show the arc adjust the slider. To show the exterior angles you have more choices, use the select control to choose the exterior angles clockwise or anticlockwise. Or the outer angle which is 360°âˆ' interior. Adjust the arc for this angle with the adjacent slider.

Styling Section

The boxes for each colour are labelled and you can also move the mouse cursor over a box for a more detailed description in the top bar. However it is important to realize that some colours are initially off/transparent.

For example the background circle is useful as students will soon realize that for regular polygons the vertices always lie on the circumference of the circle at equal intervals. The circle obviously is also associated with the radius control

The spokes of the polygon (Mathematical name?) can also be displayed. These are lines that connect the vertices to the centre point of the circle. They are useful when discussing the programming terminal. But also the angle between the spokes will match the exterior angle. The spokes be used when talking about rotational symmetry when moving the angle slider

Programming Terminal

Here at Visnos we believe that small amounts of computer programming can greatly enhance student learning opportunities. Learning a whole programming language can be daunting, but making some changes to an existing piece of code is not too difficult. Children enjoy seeing how their alterations can instantly create interesting results.

This is the first activity that includes the terminal, when it is clicked a dialogue will open up and the code(written in javascript) can be altered. Clicking the RUN button will now swap the new code and if there is no error a success message will appear in the main demo bar. If the code contains errors which is quite easy an error message will appear in the bar.

Understanding the function

To produce regular polygons the program uses a function which is passed an index which corresponds to the number of the point or vertex and the current radius as read from the radius slider control. This radius can be thought of as the spoke length. The function simply returns a number which corresponds to the length the spoke will have.

This sounds complicated but is actually quite easy to understand once you start changing the values. A video has been produced with many examples of using the terminal. A list of some of the video examples is shown below, try them out by typing them into terminal and pressing RUN. Please don't expect to understand all the code at once!

return 3;

Instead of returning the radius, the function now return 3. The polygon produced is a regular one of radius 3, and moving the radius slider no longer has any effect. If you have a background circle, notice does still change when the slider is moved.

return radius/2;

The / symbol is used by computers rather than Ã·, so radius/2 means divide the radius value each time by 2. So now another regular polygon is produced, but with a radius of half the amount shown for the radius control. Congratulations you have hacked the program to tell a lie!

return radius*2;

The * is a computers multiply symbol, so this now the radius will be double what the control says.

return radius*0.5;

So this is the same result as radius/2, but see how quickly programming allows math principles are incorporated. In this case 0.5 is the same as a half and multiplying by a half is the same as dividing by two.

return radius*index/5;

Now its starts to get more interesting, the index is included in the return value. The index starts at zero and then is increased by one for each new vertex, so this means the spoke lengths will be 0*radius/5,1*radius/5,2*radius/5 etc. This means the spoke length will get bigger for each new side. Set the sides to 100 and adjust the radius and you can see the spiral like effect produced.

if ( index == 2 )
	  return 0;
	else
	  return radius;

This is an example of a conditional statement, if the index is equal to two then zero is returned, so the spoke length will be zero. In all other cases return the radius.

if ( index % 2 == 0 )
	  return radius;
	else
	  return radius/2;

This final example includes a conditional statement which uses the % or modulus operator. The modulus operator returns the integer remainder of dividing the preceding number. So in this case index % 2 == 0 is finding the remainder from dividing by two, if the remainder is zero then the number is even. So the code is actually return the radius if the index is even or returns the radius divided by two if the index is odd.

Related activities

An interesting use of regular polygons is to use them to find an approximation of the special number π this is done pi explained activity. There are further activities for studying angles in particular basic angles.

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Interactive Polygon | math | interior angles | exterior angles | online

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Polygon explorer to learn about the properties of regular polygons. View interior angles and exterior angles. Drag vertices to create irregular polygons. Use online protractor to measure the angles.