This paper presents an original semi-analytical method (SAM) for computing the 2-D current distribution in conductors and superconductors of arbitrary shape, discretized in triangular elements. The method is a generalization of the one introduced by Brandt in 1996, and relies on new and compact analytical relationships between the current density (Jz), the vector potential (Az), and the magnetic flux density (Bx, By), for a linear variation of over 2-D triangular elements. The derivation of these new formulas, which is also presented in this paper, is based on the analytic solution of the 2-D potential integral. The results obtained with the SAM were validated successfully using COMSOL Multiphysics, a commercial package based on the finite-element method. Very good agreement was found between the two methods. The new formulas are also expected to be of great interest in the resolution of inverse problems.