Particle shape, a property of a three-dimensional solid, is an important property. Usually, and problematically, shape measurements of the solid particle are obtained only from two dimensions. It is well known that, in general, a given 2D shape measurement cannot be uniquely related to a solid. However, it is shown herein that unique, and therefore discriminating and useful, probabilistic relationships do exist between a convex solid's shape and 2D measurements taken in either section or projection. The key is to recognize that all possible 2D measurements of a solid make a characteristic probability density surface for an appropriate domain. The measurement domain studied in this paper is often used to measure actual particles. In principle a measured probability density surface of physical samples may be quantitatively approximated by mixtures of model solids. Published by Elsevier B.V.