there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon? I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use of both $\pi$ and $\sqrt{2}$ .
I would also like to know if it's possible to generalize this relatively to any n-sided polygon ( even a 3D figure ), $\sqrt{2}$ that usually appears in quadrilaterals only.
Of course I would like to know about any possible domain, but since we should start from something, I would say that the domain of polygons and polyhedron triggers my interest in the first place.

$$t_n=\bigg(-\dfrac14\bigg)^n\cdot\dfrac{\displaystyle{2n\choose n}}{1-2n}\qquad=>\qquad\sum_{n=0}^\infty t_n=\sqrt2~,\qquad\qquad\sum_{n=0}^\infty t_n^2=\dfrac4\pi$$

This is based on the fact that $~\displaystyle\sum_{n=0}^N{N\choose n}=2^N,~$ and $\quad\displaystyle\sum_{n=0}^N{N\choose n}^2={2N\choose N}.~$ Now let $N=\dfrac12$ 
and use the fact that $\Big(\tfrac12\Big)!=\dfrac{\sqrt\pi}2$ . See binomial series, Vandermonde's identity, and $\Gamma$ function 
for more details.
