# Efficiently computing the exact union of 'N' intersecting circles on a plane

I have a set of $N$ circles with a set of known radii $r_i$ on a plane of some surface area $A$. If intersection is allowed between circles, how can I most efficiently compute the exact (not approximate) surface area of their union?

See this earlier MathOverflow question: the union of circles can be constructed (as its set of boundary arcs) in $O(n\log n)$ time. Once you have that, it's straightforward to use Green's theorem to calculate the area.