The resonant frequencies of low and high order modes of square cuboid, high permittivity dielectric resonators have been studied, both theoretically and experimentally for the case of a resonator on a substrate, such as an integrated circuit. A previous analysis for an isolated rectangular resonator has been extended for the present case and Fortran IV programs have been written to calculate the resonant mode frequencies as a function of the aspect ratios of the resonators. The dimensions of resonators for various aspect ratios ranging from 0.2 to 0.9 were calculated, keeping the frequency of the fundamental TE 11¿¿mode constant at 8.2 GHz. The resonators were made from zirconium titanium stanate (¿¿r = 34.3) for the experimental determination of resonant frequencies. There was a good agreement between the theoretical and experimental values of the resonant frequencies of the various modes. It was shown from the determination of the values of Q of the various modes that the E13¿¿ mode offers a higher value of Q than the corresponding fundamental TE11¿¿ mode.