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Optimal cross-sectional shapes of whispering-gallery ring resonators with prescribed emission wavelength and resonance mode are generated using topology optimization based on the finite element method. The two critical performance indices, the quality factor (Q factor) and mode volume of a resonator, are treated as the objective functions in the optimization. In our numerical study, characteristics of optimal configurations are identified and analyzed. Since the Q factor and mode volume have a tradeoff relationship, i.e., an increasing Q factor increases mode volume, a Pareto-optimal set of solutions can be identified under certain device specifications. These configurations achieve better performances than existing shapes in producing both a high Q factor and low mode volume.