Summation of all the distances from a random point in a circle to the points on the circle?

Question

Suppose we have a circle $C$ with radius $r$ and a random point in that circle called $P$. Now if we draw lines from the point $P$ to every points on the circumference of the circle $C$ what will be summation of all the distances? My intuition is it will be $\pi r^2$ as all the lines will eventually fill the whole circle thus adding them will result the area of the circle. And will this hold for higher dimensions as well?