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

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.)

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.

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.

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