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# area of regular polygon

###### Theorem 1.

Given a regular $n$-gon with apothem of length $a$ and perimeter $P$, 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

$\begin{array}[]{rl}A&\displaystyle=n\left(\frac{1}{2}as\right)\\ &\\ &\displaystyle=\frac{1}{2}a(ns)\\ &\\ &\displaystyle=\frac{1}{2}aP.\end{array}$

∎

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.

## Mathematics Subject Classification

51-00*no label found*

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