Using modal expansions of electromagnetic fields in terms of prolate spheroidal wave functions, an exact formulation is presented for the scattering by a conducting prolate spheroid of a plane electromagnetic wave of arbitrary polarization and incidence angle. In this formulation the column vector of the series coefficients of the scattered field is obtained from the column vector of the series coefficients of the incident field by means of a matrix transformation. The matrix depends only on the scatterer; hence the scattered field for a new direction of incidence is obtained without repeatedly solving a new set of simultaneous equations. The exact curves numerically obtained for the monostatic radar cross section and relative phase of the scattered field as a function of aspect angle (angle of incidence) for both principal polarizations are given for a variety of prolate spheroids in the resonance region.