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.

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 $b = AB$ is the base of $\triangle ABH$.

Let $h = GH$ be its altitude.

See the diagram.

Then:

$\ds \triangle ABH$

$=$

$\ds \frac {b h} 2$

$\ds $

$=$

$\ds \frac b 2 \frac b 2 \cot \alpha$

Definition of Cotangent of Angle

$\ds $

$=$

$\ds \frac {b^2} 4 \cot \frac \pi n$

$\alpha$ 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 $n$ such triangles, each of which has the same area as $\triangle ABH$.

Hence:

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

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Regular Polygon of $n$ Sides Each of Length $b$: $4.9$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Regular Polygon of $n$ Sides Each of Length $b$: $7.9.$