We present a reliable numerical method that computes the spring constants of a rectangular Atomic Force Microscopy (AFM) cantilever when a small number of experimental resonance frequencies are known. By using a collection of multivariate polynomial regressions followed by an algorithm that solves a set of non-linear equations, the method finds first the values of the cantilever geometric parameters associated with the given set of resonances. The results are then used to determine the normal, longitudinal, and transversal spring constants through a second collection of regressions. Both data collections were built with finite element analysis of realistic rectangular AFM cantilevers. Experiments performed on commercial cantilevers were used to test the numerical method. The computed geometric parameters were confirmed with scanning electron microscopy, demonstrating that the method is viable, self-consistent, and accurate.