Area of Regular Polygon

A regular polygon of sides can be divided into congruent triangles by drawing radii from the center to the vertices. Each triangle has a base of times the polygon perimeter and a height of the apothem, which extends from to the midpoint of a side. Hence total area equals half the perimeter times the apothem. The apothem is always shorter than the radius , but as increases the ratio approaches . It follows that the area of a circle equals half its perimeter times ,or . For a regular polygon, the ratio of