Area of Regular Polygon

Theorem
Let $P$ be a regular $n$-sided polygon whose side length is $b$.

Then the area of $P$ is given by:
 * $\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$

where $\cot$ denotes cotangent.

Proof

 * RegularPolygonArea.png

Let $H$ be the center of the regular $n$-sided polygon $P$.

Let one of its sides be $AB$.

Consider the triangle $\triangle ABH$.

As $P$ is regular and $H$ is the center, $AH = BH$ and so $\triangle ABH$ is isosceles.

Thus $AB$ is the base of $\triangle ABH$.

Let $h = GH$ be its altitude.

See the diagram.

Then:

The full polygon $P$ is made up of $n$ such triangles, each of which has the same area as $\triangle ABH$.

Hence:
 * $\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$