What is the average distance between two random points in a regular polygon with $n$ sides? The length of each side is $l$. (Integration is not allowed).
We know:
If $n = 4$ (square), the answer is (for $l=1$) $$\frac{\sqrt2}{15}+\frac{1}{3}\ln{\mid{\sqrt{2}+1}\mid}+\frac{2}{15} = 0.52140543...$$ If $n = \infty$ (circle), the answer is $$\frac{128\cdot\text{radius}}{45\pi}$$