So I cooked up this expression for $1/\pi$, but I'm not immediately able to prove it, although computation shows it to be true. Perhaps somebody can think of a neat proof!
$\mathrm{Let}~~S_n=\{\omega \in \mathbb{C}: w^n=1\}.\quad \mathrm{Let}~~ a_n=\frac{1}{n}~\sup_{T~ \subseteq S_n}\left|\sum_{\omega \in T}\omega \right|.$
$\mathrm{Then}~~ a_n \to \frac{1}{\pi} ~\mathrm{as}~~ n \to \infty.$
(It seems that the value of $a_n$ can usually be attained by picking all of those $n^{th}$ roots of unity in the upper half plane, for example. The limit would follow from a proof of this observation.)
Thanks, and enjoy!