Experimental results from thermally annealed particles of silver and gold in the size range 10-50 nm show the presence of re-entrant surfaces on the decahedral multiply twinned particles (MTPs). A theoretical model is developed to explain these results based on the Wulff construction modified to include twin-boundaries. In addition to reproducing the re-entrant surfaces, the theory suggests that the total surface energy of the decahedral form is sandwiched between those of a single crystal and an icosahedral MTP. Consequently the stability of the decahedral particles (in a certain size range) relative to single crystals can now be explained. A useful perturbation modification is also outlined which enables qualitative comparisons of the surface energies to be made, and shows that the twin boundaries in MTPs limit the possible surface facets. Explicit shapes and energies are presented for two extreme models of the faceting. Finally, a model is outlined to justify the difference between the structures reported here and the more common decahedra and Icosahedra—the standard forms are artefacts of the growth process; they are not the thermodynamically most stable structures.