This research investigates the plane-wave scattering from a two-dimensional arbitrarily shaped cavity embedded in an infinite metallic surface that has been covered with a layer or layers of dielectric material, considering both transverse electric and transverse magnetic polarizations. Due to the shape of the cavity, this problem is approached using the finite element method. This approach provides a boundary condition at the opening of the cavity which accounts for the effect of the overlayer(s) while confining the problem to the finite domain of the cavity itself. After determination of the solution for the electric and magnetic fields at the cavity aperture, the strength of the return echo can then be calculated and displayed in a radar cross section. Id addition, numerical verifications and experiments illustrating the efficacy of the approach are provided by comparison to other previously tested methods.