In computational studies of the body mass and surface area of vertebrates, it is customary to assume that body cross-sections are approximately elliptical. However, a review of actual vertebrate cross-sections establishes that this assumption is not usually met. A new cross-sectional model using superellipses is therefore introduced, together with a scheme that allows estimates to be given with ranges. Tests of the new method, using geometrical shapes, miniature vertebrate models, and actual animals, show that the method has a high accuracy in body mass estimation. A new computer program to perform the computation is introduced. The application of the method to some Mesozoic marine reptiles suggests that the tuna-shaped ichthyosaur Stenopterygius probably had body masses comparable to those of average cetaceans of the same body length.