The sum of the radii of inscribed and circumscribed circles for an $n$ -sided regular polygon of side $a$, is:

For a regular $n$ -sided polygon with side $a$:Circumradius $R=\frac{a}{2\sin (\pi /n)}$ Inradius $r=\frac{a}{2\tan (\pi /n)}$ Sum $S=R+r=\frac{a}{2\sin (\pi /n)}+\frac{a\cos (\pi /n)}{2\sin (\pi /n)}$ $
$S=\frac{a(1+\cos (\pi /n))}{2\sin (\pi /n)}$$Usinghalf-angleformulas$ 1 + \cos\theta = 2 \cos^2(\theta/2)$and$ \sin\theta = 2 \sin(\theta/2) \cos(\theta/2)$:$$S=\frac{a(2{\cos }^2(\pi /2n))}{2(2\sin (\pi /2n)\cos (\pi /2n))}$ $
$$S=\frac{a\cos (\pi /2n)}{2\sin (\pi /2n)}=\frac{a}{2}\cot \left(\frac{\pi }{2n}\right)$$