The sum of the radii of inscribed and circumscribed circles for an...

\tan \left(\frac{\pi}{n}\right)=\frac{a}{2 r} ; \sin \left(\frac{\pi}{n}\right)=\frac{a}{2 R} r+R=\frac{a}{2}\left[\cot \frac{\pi}{n}+\operatorname{cosec} \frac{\pi}{n}\right] \Rightarrow r+R=\frac{a}{2} \cdot \cot \left(\frac{\pi}{2 n}\right)