Chapter 7.1, Problem 58E

To determine

To determine the area of the polygon of n sides of length a.

Explanation of Solution

Given information :

Length of side is $a$ .

Let us take a polygon of $n$ sides, with length of side $a$

Here $\angle AOB=\frac{{360}^{\circ }}{2n}=\frac{{180}^{\circ }}{n}$

Here OA = OB

Hence the area of triangle AOB is,

  $\frac{1}{2}\cdot \frac{a}{2}\cdot OC$

Now $OC=\frac{a}{2\tan \angle BOC}=\frac{a}{2\tan \left(\frac{{180}^{\circ }}{n}\right)}$

Now we put the values we derived, we get,

  $\begin{array}{l}A=\frac{1}{2}\cdot \frac{a}{2}\cdot \frac{a}{2\tan \left(\frac{{180}^{\circ }}{n}\right)}\\ A=\frac{1}{4}n{a}^2\cot \left(\frac{{180}^{\circ }}{n}\right)\end{array}$