We present models of the shapes of four Kuiper Belt objects (KBOs) and the Jovian Trojan asteroid (624) Hektor as ellipsoidal figures of equilibrium and Roche binaries. Our simulations select those figures of equilibrium whose light curves best match the measured rotational data. The best-fit shapes, combined with the knowledge of the spin period of the objects, provide estimates of the bulk densities of these objects. We find that the light curves of KBOs (20000) Varuna and 2003 EL61 are well matched by Jacobi triaxial ellipsoid models with bulk densities 992 and 2551 kg m-3, respectively. The light curves of (624) Hektor and KBO 2001 QG298 are well described by Roche contact-binary models with densities 2480 and 590 kg m-3, respectively. The nature of 2000 GN171 remains unclear: Roche binary and Jacobi ellipsoid fits to this KBO are equivalent but predict different densities, ~2000 and ~650 kg m-3, respectively. Our density estimates suggest a trend of increasing density with size.