The Proof of the Polygon Angle Sum Theorem

Introduction

We have learned that the angle sum of a triangle is 180^\circ. We have also learned that  the angle sum of a quadrilateral is 360^\circ. In getting the angle sum of quadrilaterals, we divided the quadrilateral into two triangles by drawing a diagonal. In this post, we use this method to find the angle sum of the pentagon and other polygons.

Let us extend the method stated above to pentagon (5-sided polygon). Clearly, we can divide the pentagon into three non-overlapping triangles by drawing two diagonals. Since each triangle has an angle sum of 180^\circ, the angle sum of a pentagon, which is composed of three triangles, is 540^\circ.

Polygon Angle Sum

Using the method above, we can see the pattern on the table below. The sum of a polygon with n sides is 180(n-2) degrees. Next, we summarize the polygon angle sum theorem and prove it.

Polygon Angle Sum Table

Theorem

The angle sum of a polygon with n sides is 180(n-2) degrees.

Proof

A polygon with n sides can be divided into n - 2 triangles. Since the angle sum of a triangle is 180^\circ, the angle sum of a polygon is 180^\circ (n-2).

Exercise

In the proof above, we have only considered the angle sum of a convex polygons. Does this theorem also hold on non-convex polygons? Explain your answer.

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