The author looks at ways of generating exact solutions for elastic membranes. Several methods are discussed using the radial expansion of a disk as an example. First, the author uses known solutions for three-dimensional plane strain deformations of a cylinder and show that these can be interpreted as membrane solutions. Second, the author shows how to generate new exact solutions for the plane strain problem using a technique in which the equilibrium equations are decoupled into two sets of equations: one for the deformation and one for the required strain energy function. Both two-and three-dimensional membrane theories are considered, and some differences are highlighted.