Förster resonance energy transfer (FRET) is a powerful tool for investigating heterogeneity in lipid bilayers. In model membrane studies, samples are frequently unilamellar vesicles with diameters of 20-200 nm. It is well-known that FRET efficiency is insensitive to vesicle curvature in uniformly mixed lipid bilayers, and consequently theoretical models for FRET typically assume a planar geometry. Here, we use a spherical harmonic expansion of the acceptor surface density to derive an analytical solution for FRET between donor and acceptor molecules distributed on the surface of a sphere. We find excellent agreement between FRET predicted from the model and FRET calculated from corresponding Monte Carlo simulations, thus validating the model. An extension of the model to the case of a non-uniform acceptor surface density (i.e., a phase-separated vesicle) reveals that FRET efficiency depends on vesicle size when acceptors partition between the coexisting phases, and approaches the efficiency of a uniformly mixed bilayer as the vesicle size decreases. We show that this is an indirect effect of constrained domain size, rather than an intrinsic effect of vesicle curvature. Surprisingly, the theoretical predictions were not borne out in experiments: we did not observe a statistically significant change in FRET efficiency in phase-separated vesicles as a function of vesicle size. We discuss factors that likely mask the vesicle size effect in extruded samples.