What does the Inclusion-Exclusion formula applied to balls look like?

If is a set of balls and | . | denotes the volume of an object, then this is the formula.


You just add and subtract more and more intersections till you run out. The same applies for surface area of spheres. If you're interested, here is a way too detailed proof.

This method is simple to understand. The problem is that if you have more than 4 balls you run into trouble. You are stuck with figuring out if all the intersections exists and what they are.

Conclusion: This method is great if you have a small number of balls or if you know you don't have any intersections of 4 or more balls.
If you do, then you're going to have to think a little harder.