The thermodynamic properties of coronal prominence cavities present a unique probe into the energy and mass budget of prominences. Using a three-dimensional morphological model, we forward model the polarization brightness and extreme-ultraviolet (EUV) emission of a cavity and its surrounding streamer. Using a genetic algorithm, we find the best-fit density model by comparing the models to Mauna Loa Solar Observatory MK4 and Hinode EUV Imaging Spectrometer data. The effect of temperature variations on the derived density is also measured. We have measured the density inside a cavity down to 1.05 R sun with height-dependent error bars. Our forward modeling technique compensates for optically thin projection effects. This method provides a complementary technique to traditional line ratio diagnostics that is useful for diffuse off-limb coronal structures.