This is just out of the top of my head so could be wrong but isnt't there a simple analytic solution by using Riemann sums: Pick any point in the intersection

This is just out of the top of my head so could be wrong but isnt't there a simple analytic solution by using Riemann sums: Pick any point in the intersection A (can be on border too). Divide the interval from 0 to 2pi into many small chunks of length delta-theta. 1/2 * Sum for n from 0 to 2pi in delta-theta chunks of min(distance from A to each circle in n angle direction)*delta-theta should be the area of intersection.