# Is there a formula for the area of a regular polygon of side a having n sides?

Nov 30, 2016

$S = \frac{n {a}^{2}}{4} \cot \left(\frac{\pi}{n}\right)$

#### Explanation:

A regular polygon has all sides equal $a$. Each side is observed from the center $O$ at the same angle $\phi$. The polygon area is covered by as many isosceles triangles with vertices centered, as sides it has. A regular polygon has a circumscribed circle with radius $r$. Considering now the radius $r$ as the side of isosceles triangle with vertice at the center and $a$ as third side we have the area $s$

$s = \frac{1}{2} a r \cos \left(\frac{\phi}{2}\right)$

if we have $n$ sides then $\phi = \frac{2 \pi}{n}$

and the polygon area is given by

$S = n s = \frac{n a r}{2} \cos \left(\frac{\pi}{n}\right)$

but $a = 2 r \sin \left(\frac{\phi}{2}\right) = 2 r \sin \left(\frac{\pi}{n}\right)$

now substituting $r = \frac{a}{2 \sin \left(\frac{\pi}{n}\right)}$ into the $S$ relationship

$S = \frac{n {a}^{2}}{4} \cot \left(\frac{\pi}{n}\right)$