We investigate the problem of inferring elastic moduli of nonlinearly elastic membranes from interior measurements of deformation and pressure. We begin by formulating a model of membrane deformation under a vertical force where the geometry of the membrane is star-like. The model makes no specification of the constitutive law by which stresses are calculated from applied strains. Under appropriate choice of the constitutive law and simplification of the geometry, we show that membranes of regular structure may be homogenized to an axisymmetric case. We then investigate numerical methods for the resolution of the axisymmetric model in terms of radial and vertical displacement. Examples are given for various boundary conditions and choices for elastic moduli. We present a method by which the moduli may be accurately recovered by algebraic calculation from knowledge of the displacements on the boundary and interior of the membrane together with measurement of the radial stress at one of the boundaries.