Given a surface with area $A$, what is the expected area of the region occupied by $k$ possibly overlapping random circles with equal radii $r$?
For example, I would like to estimate the area of the black region of this simulation.
If it helps, we can assume the edges of surface wraps around, so the surface is a torus. And instead of random circles, it can also be random squares.
I've tried to find an approximation by partitioning the space as a "pixel" grid and solve it as a K balls in N buckets problem. But I didn't find how to generalize it from balls to circles.