A regular polygon of [math]N[/math] sides can be divided into [math]N[/math] congruent triangles by drawing radii from the center [math]O[/math] to the vertices. Each triangle has a base of [math]1[/math]/[math]N[/math] times the polygon perimeter and a height of the apothem, which extends from [math]O[/math] to the midpoint of a side. Hence total area equals half the perimeter times the apothem. The apothem is always shorter than the radius [math]r[/math], but as [math]N[/math] increases the ratio approaches [math]1[/math]. It follows that the area of a circle equals half its perimeter [math]2\pi r[/math] times [math]r[/math], or [math]\pi r^2[/math].

A regular polygon of [math]N[/math] sides can be divided into [math]N[/math] congruent triangles by drawing radii from the center [math]O[/math] to the vertices. Each triangle has a base of [math]1[/math]/[math]N[/math] times the polygon perimeter and a height of the apothem, which extends from [math]O[/math] to the midpoint of a side. Hence total area equals half the perimeter times the apothem. The apothem is always shorter than the radius [math]r[/math], but as [math]N[/math] increases the ratio approaches [math]1[/math]. It follows that the area of a circle equals half its perimeter [math]2\pi r[/math] times [math]r[/math], or [math]\pi r^2[/math].