Area of a Regular Polygon

This proof will be done for a pentagon but holds true for any other regular polygon. Consider a pentagon with apothem a and side length l.

This pentagon can be divided into 5 triangles. Since this is a regular pentagon, its sides are congruent.

A circumscribed circle can be drawn around this pentagon. The remaining sides of the triangles are congruent since they are radii of the circumscribed circle.

Therefore, the five triangles inside the pentagon are congruent by the Side-Side-Side Congruence Theorem. Also, each apothem is perpendicular to its associated side, so each triangle has base l and height a.

Using the formula for the area of a triangle, the area of each triangle is 12l a. The pentagon is divided into five congruent triangles, so the area of the pentagon is the sum of the areas of those five triangles. A = 5 * 1/2l a ⇓ A = 1/2 (5l)* a Since 5l is the perimeter p of the pentagon, the following formula can be derived.