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.

Stirling's approximation:
$$
n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n
$$
