Circles of a single size can pack together densely in a hexagonal lattice, but adding in size variety disrupts the order of those packings. We conduct simulations which generate dense random packings of circles with specified size distributions and measure the area fraction in each case. While the size distributions can be arbitrary, we find that for a wide range of size distributions the random close-packing area fraction phi_rcp under this general protocol is determined to high accuracy by the polydispersity and skewness of the size distribution. At low skewness, all packings tend to a minimum packing fraction phi0 ≈ 0.840 independent of polydispersity. In the limit of high skewness, phi_rcp becomes independent of skewness, asymptoting to a polydispersity-dependent limit. We show how these results can be predicted from the behavior of bidisperse or bi-Gaussian circle size distributions.