With some number of points that are evenly/uniformly (assuming those mean the same thing) distributed within a circle of radius 1, what is the average distance from the center of the circle to a point?
Is it something like 0.707 (sqrt(2)/2) or something like 0.666 (2/3)?
If we put a uniform distribution over a unit circle, then the pdf of the distance $\rho$ from the center is given by: $$ f_\rho(x) = 2x\cdot\mathbb{1}_{[0,1]}(x) \tag{1}$$ hence the average distance is given by: $$ \int_{0}^{1} 2x^2\,dx = \color{red}{\frac{2}{3}}.\tag{2}$$
Consider the solid obtained by deleting from the cylinder with base a unit circle and unit height, the inverted right cone with the base being the top of the cylinder. Note that the height of this solid over any point $x$ in the unit circle base is exactly the distance between $x$ and the center. So the required average is \begin{eqnarray}\frac{\text{Volume of the solid}}{\text{Volume of the cylinder}}\end{eqnarray} which is $\displaystyle 1-\frac{1}{3}=\frac{2}{3}$.
