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Math Monday (By Special Request) Internal Angles In Polygons
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Math Monday (By Special Request) Internal Angles In Polygons

Can you believe that there has actually been a request for a nerdy math post here on CTP Teams? Look I will show you.

Sunny CTP Teams Question

It came from Sunny Suggs after I referred to the sum of the internal angles of a 10-sided polygon in this blog post – What Does 1440 Mean To You?

So let’s get nerdy with some math. Feel free to walk away now if this is of no interest to you because there is nothing but math in this post, although not difficult math.

Let’s take the starting point that the interior angles of a triangle (any triangle) will always add up to 180 degrees. This means that in any given triangle only one of the angles can 90 degrees or greater and in a right angled triangle (where one angle is always 90 degrees) the other two angles must each be 45 degrees.

For some triangle fun – and to prove this for yourself then click on the image below and have a little play with the graphic to move the triangle and see what happens to its angles. (It will open up in a new tab so don’t forget to come back here when you are done playing.)

Interior angles of a triangle
Next up, a four sided shape or quadrilateral (such as a square or rectangle) is able to be cut in to two triangles so it follows that its internal angles must add up to 360 degrees, (180 + 180 degrees.)

In addition if the quadrilateral is one where all four vertices lie on a circle, known as a cyclic or inscribed quadrilateral, then each pair of opposite interior angles will always add up to 180 degrees. Again, click on the image below to find out for yourself.

interior angles of a quadrilateral

A five-sided shape, called a pentagon, can be cut into three triangles so its interior angles must add up to 3 x 180 = 540 degrees. A six sided polygon, a hexagon, can be cut into four triangles so its interior angles must add up to 4 x 180 = 720 degrees.

So from this pattern we can work out a formula to find the sum of the internal angles of any polygon, thus:

where the number of sides on the polygon = n then the sum of that polygon’s internal angles will be sum = 180 x (n-2) degrees.

So the sum of the internal angles of a 10-sided polygon must be

180 x (10-2) degrees

= 180 x 8 degrees

= 1440 degrees which is the figure given in the original blog post.

Finally if you want to play with some more polygons then click on this link below.

Sum of internal angles in any polygon

All graphic images used in this post are from Math Open Reference.

10 responses to “Math Monday (By Special Request) Internal Angles In Polygons”

  1. Tom Wacker says:

    Well. I certainly hope that Sunny got the information that she needed from this post. Actually, a very cool post, and the playstations are really cool.

    Thanks Patrick!

  2. I always liked math simple algebra is fun goooo Sunny

  3. Matt Badura says:

    Never too old for a refresher 😉 Thanks Patrick

  4. Kathy Dyer says:

    Cool post and very interesting!

  5. “A long time ago in a galaxy far, far away…” Miss Jacobson’s Algebra III (trigonometry) class 😀

  6. Marye says:

    Use it or lose it…I definitely lost it…lol

  7. Sunny Suggs says:

    Yay! thank you Patrick! I had mucho fun playing with the shapes and angles.

  8. Karen Kuty says:

    Thanks, Patrick.

    The inner nerd in me is purring away…

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