Controlling the geometry of self-assembly will enable a greater diversity of nanoparticles than now available. Viral capsid proteins, one starting point for investigating self-assembly, have evolved to form regular particles. The polyomavirus SV40 assembles from pentameric subunits and can encapsidate anionic cargos. On short ssRNA (≤814 nt), SV40 pentamers form 22-nm-diameter capsids. On RNA too long to fit a T=1 particle, pentamers forms strings of 22-nm particles and heterogeneous particles of 29 to 40 nm diameter. However, on dsDNA SV40 forms 50 nm particles composed of 72 pentamers. A 7.2-Å resolution cryo-EM image reconstruction of 22-nm particles shows that they are built of twelve pentamers arranged with T=1 icosahedral symmetry. At threefold vertices, pentamers each contribute to a three-helix triangle. This geometry of interaction is not seen in crystal structures of T=7 viruses and provides a structural basis for the smaller capsids. We propose that the heterogeneous particles are actually mosaics formed by combining different geometries of interaction from T=1 capsids and virions. Assembly can be trapped in novel conformations because SV40 interpentamer contacts are relatively strong. The implication is that by virtue of their large catalog of interactions, SV40 pentamers have the ability to self-assemble on and conform to a broad range of shapes.