The wrong way of finding the average distance between two points on a circle I was trying to find the average distance between two points on a circle and got the following result. 
Why is my method wrong?

Let $P_1$ be fixed while $P_2$ moves around a circle. The distance between these points is:
$$s=2R\sin{\alpha \over 2}$$
...where $\alpha$ represents the central angle corresponding to points $P_1,P_2$
Because of symmetry we can check only one half of the circle to calculate the average distance:
$$d=\frac{\int sdl}{\int dl}$$
$$d=\frac{\int_0^\pi 2R\sin\frac\alpha2\cdot Rd\alpha}{R\pi}$$
$$d=\frac{-4R^2cos\frac\alpha2|_0^\pi}{R\pi}=\frac{4R}{\pi}$$
