area of regular polygon (original) (raw)

Proof.

Given a regular n-gon R, line segments can be drawn from its center to each of its vertices. This divides R into n congruent triangles. The area of each of these triangles is 12⁢a⁢s, where s is the length of one of the sides of the triangle. Also note that the perimeter of R is P=n⁢s. Thus, the area A of R is

A=n⁢(12⁢a⁢s)=12⁢a⁢(n⁢s)=12⁢a⁢P.

∎

To illustrate what is going on in the proof, a regular hexagon appears below with each line segment from its center to one of its vertices drawn in red and one of its apothems drawn in blue.