A mean-field theory for the geometry and diffusive growth rate of soap bubbles in dry 3D foams is presented. Idealized foam cells called isotropic Plateau polyhedra (IPPs), with F identical spherical-cap faces, are introduced. The geometric properties (e.g., surface area S, curvature R, edge length L, volume V) and growth rate of the cells are obtained as analytical functions of F, the sole variable. IPPs accurately represent average foam bubble geometry for arbitrary F ≥ 4, even though they are only constructible for F = 4,6,12. While R/V1/3, L/V1/3 and exhibit F1/2 behavior, the specific surface area S/V2/3 is virtually independent of F. The results are contrasted with those for convex isotropic polyhedra with flat faces.