"The number $\frac 16 \pi^2$ turns up surprisingly often and frequently in unexpected places." - Julian Havil,

Gamma: Exploring Euler's Constant.

It is well-known, especially in 'pop math,' that
$$\zeta(2)=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\cdots = \frac{\pi^2}{6}.$$
Euler's proof of which is nice. **I would like to know where else this constant appears non-trivially.** This is a bit broad, so here are the specifics of my question:

- We can fiddle with the zeta function at arbitrary even integer values to eek out a $\zeta(2)$. I would consider these 'appearances' of $\frac 16 \pi^2$ to be redundant and ask that they not be mentioned unless you have some wickedly compelling reason to include it.
- By 'non-trivially,' I mean that I do not want converging series, integrals, etc. where it is obvious that $c\pi$ or $c\pi^2$ with $c \in \mathbb{Q}$ can simply be 'factored out' in some way such that it looks like $c\pi^2$ was included after-the-fact so that said series, integral, etc. would equal $\frac 16 \pi^2$. For instance, $\sum \frac{\pi^2}{6\cdot2^n} = \frac 16 \pi^2$, but clearly the appearance of $\frac 16\pi^2$ here is contrived. (But, if you have an answer that seems very interesting but you're unsure if it fits the 'non-trivial' bill, keep in mind that nobody will actually stop you from posting it.)

I hope this is specific enough. **This was my attempt at formally saying 'I want to see all the interesting ways we can make $\frac 16 \pi^2$.'** With all that being said, I will give my favorite example as an answer below! :$)$

There used to be a chunk of text explaining why this question should be reopened here. It was reopened, so I removed it.