We provide an algorithm that computes algebraic quadrature formulas with cardinality not exceeding the dimension of the exactness polynomial space, on the intersection of any number of planar disks with arbitrary radius. Applications arise for example in computational optics and in wireless networks analysis. By the inclusion-exclusion principle, we can also compute algebraic formulas for the union of a small number of disks. The algorithm is implemented in Matlab, via subperiodic trigonometric Gaussian quadrature and compression of discrete measures.