## What is the sum of the angles of an octagon?

There's a simple way to calculate the sum of all the interior angles of an octagon, or any polygon. To see how it works, first draw yourself an octagon (it doesn't have to be perfect). Now, pick any corner of that octagon, and then draw a line from each of the other corners of the octagon to that corner. (You won't draw lines from the corners closest to the one you chose because those lines are already there — they're two sides of your octagon!)

Now take a look at what you have. You should see six non-overlapping triangles stuck together to make an octagon. Notice how every angle in each of those triangles is part of one of the angles of the octagon. That means that if you add up all the angles in those six triangles, you'll get the total internal angle sum of the octagon.

In this case, 6 x 180 = 1080; an octagon's internal angles add up to 1080 degrees.

You can do this with any convex polygon, and by *convex*, I mean that all the internal angles are less than 180 degrees. If you do a little investigating, you'll find that the number of triangles is always two less than the number of sides. This is so regular that it's stated as a theorem:

If a convex polygon has *n* sides, then its interior angle sum (*S*) is given by the following equation:

*S* = ( *n* – 2) × 180°

With this equation, you can calculate the interior angle sum of polygons with 37 sides (6300 degrees), 73 sides (12,780 degrees), or even 100 sides (17,640 degrees) without knowing any other information.