Lifetimes are estimated for the two-dimensional resonant optical modes of a dielectric cavity with an equilateral triangular cross section, that are approximately confined by total internal reflection. Exact solutions of a two-dimensional scalar wave equation for triangular geometry with Dirichlet boundary conditions are used to describe approximately the vector fields of the possible transverse electric (TE) and transverse magnetic (TM) modes. Only two-dimensional electromagnetic solutions are considered here, where there is no propagation vector perpendicular to the plane of the triangle (k_z = 0). The field properties just inside and outside the cavity boundary are shown to be significantly different for TE and TM field polarizations, the two cases having different dependences on the index mismatch with the exterior. For a given mode specified by particular quantum numbers, TE polarization leads to longer lifetime than TM polarization at high index mismatch, assuming that escape of evanescent boundary waves at the corners is the primary decay process.