# 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

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:

\(\displaystyle \triangle ABH\) | \(=\) | \(\displaystyle \frac {b h} 2\) | Area of Triangle in Terms of Side and Altitude | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac b 2 \frac b 2 \cot \alpha\) | Definition of Cotangent of Angle | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {b^2} 4 \cot \frac \pi n\) | $\alpha$ is half the apex of $\triangle ABH$, and $n$ of such apices fit into the full circle of $2 \pi$ |

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

Hence:

- $\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 4$: Geometric Formulas: $4.9$