Area of Regular Polygon
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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:
\(\ds \triangle ABH\) | \(=\) | \(\ds \frac {b h} 2\) | Area of Triangle in Terms of Side and Altitude | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac b 2 \frac b 2 \cot \alpha\) | 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:
- $\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: $4.9$