A simple non-Maxwellian method is presented that allows the approximate solution of all the dimensions of a multilayered dielectric TE0qp mode cylindrical resonant cavity that constitutes a distributed Bragg reflection (DBR) resonator. The analysis considers an arbitrary number of alternating dielectric and free-space layers of cylindrical geometry enclosed by a metal cylinder. The layers may be arranged along the axial direction, the radial direction, or both. Given only the aspect ratio of the cavity, the desired frequency and the dielectric constants of the material layers, the relevant dimensions are determined from only a set of simultaneous equations, and iterative techniques are not required. The formulas were verified using rigorous method of lines (MoL) calculations and previously published experimental work. We show that the simple approximation gives dimensions close to the values of the optimum Bragg reflection condition determined by the rigorous analysis. The resulting solution is more compact with a higher Q-factor when compared to other reported cylindrical DBR structures. This is because it properly takes into account the effect of the aspect ratio on the Bragg antiresonance condition along the z-axis of the resonator. Previous analyses assumed the propagation in the z-direction was independent of the aspect ratio, and the layers of the Bragg reflector were a quarter of a wavelength thick along the z-direction. When the aspect ratio is properly taken into account, we show that the thickness of the Bragg reflectors are equivalent to the thickness of plane wave Bragg reflectors (or quarter wavelength plates). Thus it turns out that the sizes of the reflectors are related to the free-space propagation constant rather than the propagation constant in the z-direction.