To investigate how the curvature of a reactive surface can affect reaction kinetics, we use a simple model in which a diffusion-limited bimolecular reaction occurs on a curved surface that is hollowed inward, flat, or extended outward while keeping the reactive area on the surface constant. By numerically solving the diffusion equation for this model using the finite element method, we find that the rate constant is a non-linear function of the surface curvature and that there is an optimal curvature providing the maximum value of the rate constant, which indicates that a spherical reactant whose entire surface is reactive (a uniformly reactive sphere) is not the most reactive species for a given reactive surface area. We discuss how this result arises from the interplay between two opposing effects: the exposedness of the reactive area to its partner reactants, which causes the rate constant to increase as the curvature increases, and the competition occurring on the reactive surface, which decreases the rate constant. This study helps us to understand the role of curvature in surface reactions and allows us to rationally design reactants that provide a high reaction rate.