Interesting and unexpected applications of $\pi$ $\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$
Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
and the generalization of $\zeta (2k)$, my perception of $\pi$ has changed. I used to think of it as rather obscure and purely geometric (applying to circles and such), but it seems that is not the case since it pops up in things like this which have no known geometric connection as far as I know. What are some other cases of $\pi$ popping up in unexpected places, and is there an underlying geometric explanation for its appearance?
In other words, what are some examples of $\pi$ popping up in places we wouldn't expect?

This is too long for a comment, but this regards the $\frac{\pi}{4}$ series mentioned earlier.
Consider a unit square and a quarter of the unit circle. Cut the square down the diagonal; half of the arc of that circle will equal $\frac{\pi}{4}$. Split $AB$ into $n$ equal pieces. Then it can be shown that $$\lim_{n \to \infty} \displaystyle \sum_{r = 1}^{n} \frac{\frac1n}{1 + (\frac rn)^2} = \frac{\pi}{4}$$
A full explanation is beautifully done in a comment here.
So yes, geometric explanations exist. Circles appear everywhere in mathematics, it's almost scary.
