Geometries, energies and magnetic shieldings are reported at the ab initio B3LYP/6-31G* level for the phenylene cluster C-120 (archimedene) and eight phenylene-based hydrocarbon bowls consisting of four-, six-, and ten-membered rings. The six-membered rings are branched, angular, or terminal. The latter are the most aromatic, based upon NICS criteria and lack of substantial bond alternation. At the other extreme are branched rings, having less negative NICS values. Four-membered rings, except those on a rim, are nearly square. Regularities are found in the ab initio energies, heats of formation (Delta H degrees(f)), and strain energies relative to those of hypothetical planar acyclic analogues. The bowls appear to have little aromatic character, and their interiors are but slightly shielded. Archimedene, with Delta H degrees(f) = 2191 kcal/mol, has energetic and structural properties akin to those of phenylene-based bowls.