Question

If $R,r$ are the radii of the circumcircle and incircle of a regual polygon of $n$ sides, each being of length $a$, then $2(R+r)\tan \left(\frac{\pi }{2^n}\right)=ka$ where $k=$

Answer 1

Since $R=\frac{1}{2}a\csc \frac{\pi }{n},r=\frac{1}{2}a\cot \frac{\pi }{n}$

$\therefore 2(R+r)=a[\frac{1}{\sin \frac{\pi }{n}}+\frac{\cos \left(\frac{\pi }{n}\right)}{\sin \left(\frac{\pi }{n}\right)}]$

$=a\frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}=a.\frac{2{\cos }^2\frac{\pi }{n}}{2\sin \frac{\pi }{n}\cos \frac{\pi }{n}}$

$=a\cot \frac{\pi }{2n}$

$\therefore a=2(R+r)\tan \frac{\pi }{2n}.$

$\therefore k=1.$