We have generated off-lattice self-avoiding random walks (SAWs) by both Lal's pivot algorithm and by discontinuous molecular dynamics (DMD) simulations. We chose several volume measures to analyze and compare the shapes characterizing ensembles of SAWs. These included the Flory volume defined through the end-to-end distance, the volume corresponding to the radius of gyration of the polymer, the volume of a sphere drawn from the center of polymer mass that encloses or embeds all the monomers in the polymer, the corresponding ellipsoid describing the radius of gyration tensor, the volume of the smallest Cartesian box oriented in a fixed "lab" frame which encloses all polymer monomers, and the volume of the smallest box oriented along the principal axes of the radius of gyration tensor that encloses all polymer monomers. The tensor ellipsoid and principal box correlate well with each other but not with the other measures. There is a substantial amount of polymer that is excluded from the radius of gyration sphere or ellipsoid (approximately 42% and 44% respectively on average), which casts doubt on the utility of these measures in quantitatively characterizing the volume spanned by a polymer configuration. The principal box volume led to the most well-defined length distribution for. the SAW in terms of the ratio of standard deviation to the mean, while the end-to-end distribution was the most broadly distributed; In. the principal box analysis, polymer configurations are highly anisotropic, with stronger cubic to square prism symmetry breaking than square prism to rectangular cuboid. We introduce the acubicity in analogy with the asphericity and find its distribution, along with a generalization of acubicity that better discriminates polymer anisotropy. We analyze the role of the above volume measures in determining the packing fraction which enters into the "random mixing" mean-field theory for the energy of an isolated homopolymer. Here we find from both pivot and DMD simulations that no particular volume measure reproduces the mean-field scaling exponent (of unity) for the energy as a function of the reciprocal polymer volume. Instead, anomalous exponents are observed which are less than that of mean-field theory, ranging from 0.2 to 0.5 depending on the contact cutoff length. An energy function with such an anomalous scaling may be used in a simple phenomenological theory of coil-globule collapse. We find in particular that the Flory volume does not accurately describe the energy in a mean-field theory. Possible reasons for such anomalous exponents are discussed.