area of regular polygon

Theorem 1.

Given a regular $n$-gon (http://planetmath.org/RegularPolygon) with apothem of length $a$ and perimeter (http://planetmath.org/Perimeter2) $P$, its area is

$$A=\frac{1}{2}aP.$$

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 ${\displaystyle \frac{1}{2}}as$, where $s$ is the length of one of the sides of the triangle. Also note that the perimeter of $R$ is $P=ns$. Thus, the area $A$ of $R$ is

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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.