Given $n$ circles of radius $r$, arranged so that line segments connecting the centers from one circle to the next form a convex polygon and none of the circles intersect, the length of a string stretched tightly around the outside of all the circles is:
\begin{equation} length = 2 \pi r + \sum_{i=0}^{n-1} dist(i, (i+1)_n)\end{equation} where $dist(i, j)$ is the distance between the centers of circles $i$ and $j$.
It's easy to see this for $n=1$ and $n=2$, but for $n > 2$ my intuition fails me and it seems like it would be a pretty hairy proof. Is there a fairly straightforward explanation or is it very involved? Thanks for any pointers.