# Length of string stretched around circles of identical radius

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:

$$length = 2 \pi r + \sum_{i=0}^{n-1} dist(i, (i+1)_n)$$ 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.

-

Consider two adjacent circles with centres $C_1$ and $C_2$. Let $AB$ be the straight stretch of string between the first circle and the second. Then $C_1ABC_2$ is a rectangle with sides $r$ and $d$, where $d$ is the distance between $C_1$ and $C_2$. (A sketch is helpful here.) The straight bits of string therefore have a total length equal to the sum of the distances between the centres of adjacent circles.