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