A regular polygon of $n$ sides is formed by bending a wire of length $2\pi R$ which carries a current $i.$ The magnetic field at the Centre of the polygon $-$ 

One side of polygon $a=\frac{2\pi R}{n}$

   $\alpha =\beta =\left(\frac{2\pi }{n}\right)/2=\frac{\pi }{n}; \cot \alpha =\frac{d}{a/2}$

$d=\frac{a}{2}\cot \alpha =\frac{\pi R}{n}\cot \left(\frac{\pi }{n}\right)$

All sides of polygon produce magnetic field at $O$ in same direction

So,  $B=n(magnetic field due to one side)$

$B=n·\frac{{\mu }_{0}i}{4\pi d}(\sin \alpha +\sin \beta )$

$B=n·\frac{{\mu }_{0}i}{4\pi }\frac{n}{\pi R\cot \frac{\pi }{n}}\left(\sin \frac{\pi }{n}+\sin \frac{\pi }{n}\right)$

$B=\frac{{\mu }_{0}i}{4\pi R}\frac{n^2}{{\pi }^2}\frac{2\sin \pi /n}{\cos \pi /n}·\sin \frac{\pi }{n}$

$B=\frac{{\mu }_{0}i}{2R}\frac{\sin \pi /n/\pi /n}{\cos \pi /n}$