Analytical expressions for the resonant frequencies of an electromagnetic rectangular/cylindrical/spherical cavity with an off-axis electrically small lossless dielectric sphere are obtained in a very simple and immediate manner, for both TE and TM modes. The walls of the cavities are perfectly conducting. Rectangular/cylindrical/spherical vector wave functions are used in our analysis. The results are useful for cavities containing small inhomogeneities, as well as for the determination of permittivities or permeabilities of various materials, from the measurement of the resonant frequency shifts caused by introducing small samples of them inside cavities.