Förster resonance energy transfer (FRET) has become an important tool to study the submicrometer distribution of proteins and lipids in membranes. Although resolving the two-dimensional distribution of fluorophores from FRET is generally underdetermined, a forward approach can be used to determine characteristic FRET "signatures" for interesting classes of microdomain organizations. As a first step toward this goal, we use a stochastic Monte Carlo approach to characterize FRET in the case of molecules randomly distributed within disk-shaped domains. We find that when donors and acceptors are confined within domains, FRET depends very generally on the density of acceptors within domains. An implication of this result is that two domain populations with the same acceptor density cannot be distinguished by this FRET approach even if the domains have different diameters or different numbers of molecules. In contrast, both the domain diameter and molecule number can be resolved by combining this approach with a segregation approach that measures FRET between donors confined in domains and acceptors localized outside domains. These findings delimit where the inverse problem is tractable for this class of distributions and reframe ways FRET can be used to characterize the structure of microdomains such as lipid rafts.