Sum of N-gon Angles

Following is the derivation of the formula for determining the number of angle degrees in an n-gon (a polygon of "n" sides). This can also be found on the October Thoughts blog.

All n-gons can be broken down into triangles by creating line segments from a single angle to the points located at other angles (Theorem 3.25). The number of triangles formed can be determined by identifying the number of interior angles and subtracting the two angles located next to the chosen angle (line segments from the chosen angle to these angles would trace the two sides of the chosen angle). The number of sides on an n-gon equals the number of angles; so you can also use the number of sides less two. In other words, the number of interior triangles that can be formed in any n-gon equals the number of sides on the n-gon less two sides.

According to Theorem 4.53, all of the interior angles of a triangle sum to a linear angle which, as shown in earlier problems, is 180. The measure of the angles can be determined by multiplying 180 (per triangle) times the number of sides on the n-gon less two (number of interior triangles). Therefore, if n is the number of sides the following applies:

Number of angle degrees = (n-2)(180)