Question

The sum of inradius and circumradius of incircle and circumcircle of a regular polygon of $n$ sides and each of length $a$ is

• A. $\frac{a}{4}\cot \frac{\pi }{2n}$
• B. $a\cot \frac{\pi }{n}$
• C. $\frac{a}{2}\cot \frac{\pi }{2n}$
• D. $a\cot \frac{\pi }{2n}$

Answer 1

The correct option is C $\frac{a}{2}\cot \frac{\pi }{2n}$

Let $r$ and $R$ be the inradius and cirumradius of incircle and and circumcircle of the given regular polygon of side $n.$ Then, we know that,
Inradius of polygon $r=\frac{a}{2}\cot \frac{\pi }{n}$
Circumradius of polygon$R=\frac{a}{2}\text{cosec}\frac{\pi }{n}$
Adding values of $r+R$
We get,
$r+R=\frac{a}{2}\cot \frac{\pi }{n}+\frac{a}{2}\text{cosec}\frac{\pi }{n}$
$\Rightarrow \frac{a}{2}(\frac{\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}+\frac{1}{\sin \frac{\pi }{n}})$
$\Rightarrow \frac{a}{2}\left(\frac{\cos \frac{\pi }{n}+1}{2\sin \frac{\pi }{2n}\cos \frac{\pi }{2n}}\right)$
$\Rightarrow \frac{a}{2}\left(\frac{2{\cos }^2\frac{\pi }{2n}}{2\sin \frac{\pi }{2n}\cos \frac{\pi }{2n}}\right)$
$\Rightarrow \frac{a}{2}\cot \frac{\pi }{2n}$
$\therefore r+R=\frac{a}{2}\cot \frac{\pi }{2n}$