The equilibrium shape and dihedral angles at the solid-liquid-vapor tri-junctions of two-phase alloy small particles containing a cusp-oriented interface were modeled as a function of phase fraction, surface energy and the interfacial energy. The calculation was applied to different combinations of surface and/or interfacial energies to demonstrate the various possible particle shapes and dihedral angles that result for two-phase particles. The dihedral angles at the tri-junction vary with the phase fraction, due to the coupling between the relative amounts of each phase, interfacial energy relative to the two surface energies and the equilibrium conditions at the tri-junction. These features can be used to find the ratio of the interfacial energy to the surface energies of two-phase particles for any state of matter.