This extends to higher dimensions and other geometries. Any part of the disk is equally likely to contain a random point. Normalized probability density function is dP/dS = 1/(Pi * R^2). As dS = r * dr * du and radius r is independent of the polar angle u, we can write the density as a product of two constant densities: dP/d(r^2) * dP/du = 1/R^2 * 1/(2 * Pi). So r^2 := x * R^2, and u := x * 2 * Pi, where x is a uniform variate in [0, 1).