Combined area of overlapping circles

Question

I recently came across a problem where I had four circles (midpoints and radius) and had to calculate the area of the union of these circles.

Example image:

For two circles it's quite easy,

I can just calculate the fraction of the each circles area that is not within the triangles and then calculate the area of the triangles.

But is there a clever algorithm I can use when there is more than two circles?

Answer 1

I found this link which may be useful. There does not seem to be a definitive answer though. Google answers. Another reference for three circles is Haruki's theorem. There is a paper there as well.