Question

Find the sum of the radii of the circles, which are respectively , inscribed and circumscribed about a polygon of $n$ sides, whose side length is $a$

Answer 1

we have

Radius of the circumscribed circle $=R=\frac{a}{2}co\sec \frac{\pi }{n}$
and

Radius of the inscribed circle
$=r=\frac{1}{2}a\cot \frac{\pi }{n}$

adding

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