Unit Circle Average Distance

Let the point on the unit circle be $\displaystyle \left(\cos\theta,\sin\theta\right)$

Now, the average distance from the point $\displaystyle \left(1,0\right)$ is

$\frac{1}{2\pi}\int_0^{2\pi}\sqrt{2-2\cos\theta}d\theta$

$=\frac{1}{\pi}\int_0^{2\pi}\left|\sin\left(\frac{\theta}{2}\right)\right|d\theta$

$=\frac{4}{\pi}$