Question

Find the sum of radii of inscribed and circumscribed circles of n side regular polygon of side $\alpha$?

Answer 1

If $R$ be the radius of circumcircle of regular polygon of $n$ sides, and $r$ be the radius of inscribed circle, then
$R=\frac{a}{2}\csc \frac{\pi }{n}andr=\frac{a}{2}\cot \frac{\pi }{n}$
$\therefore R+r=\frac{a}{2}(\csc \frac{\pi }{n}+\frac{a}{2}\cot \frac{\pi }{n})$

solving trigonometric equation
$=\frac{a}{2}\left(\frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}\right)=\frac{a}{2}\cot \frac{\pi }{2n}$