A semi-analytical study of the acoustic radiation losses associated with various transverse vibration modes of a micromechanical (MEMS) annular resonator is presented. The quality factor, Q, of such resonators is of interest in many applications and depends on structural geometry, fluid-structure interaction, and the device encapsulation. Resonators with at least one surface exposed to air can display significant losses through acoustic radiation even at μm dimensions. In this study, well established analytical techniques for modeling resonator vibration modes and fluid-structure interaction are used in conjunction with numerical and symbolic computation to establish a semi-analytical procedure for computing Q due to acoustic radiation losses, Qac, in any vibrational mode. The procedure includes calculation of the exact modeshape using computer algebra with an FEM based calculation as the starting point. This modeshape is then used for acoustic field calculations that lead to the computation of Qac. The computed values of Q ac show a non-monotonic variation with the increasing modal frequencies. This non-monotonic variation in Qac is attributed to the radiation efficiency associated with various modes of the annular resonator. There exists an inverse relationship between the acoustic radiation efficiency and the Qac for various resonant frequencies. With the increase in the number of nodal circles and diameters, the radiation efficiency decreases and hence Qac increases. Results are compared for the lowest two modes of a solid circular resonator using exact mode shapes to those of Lamb's approximate mode shapes. Comparison to published experimental results validates the predictive utility of the proposed technique, especially for higher modes where acoustic radiation is the dominant constituent of Q. The hybrid analytical-computational procedure established here can be used in choosing appropriate mode shapes of the resonator for the desired Q.