Our ability to accurately quantify the total ice volume in glaciers and the loss of glacier volume, discharge and freshwater in response to climate change is limited by a paucity of ice thickness and bed topography observations. Consequently, glacial ice thickness is often inferred indirectly from more easily obtained surface measurements. Here, we present a simple inversion building on the assumption of perfect plasticity. In the traditional perfect-plastic approximation, the ice thickness (or bed) can be inferred from the surface elevation and yield strength. Here, we extend this to demonstrate that, provided glaciers are changing, we can simultaneously determine the yield strength and bed topography from observations of surface elevation alone. We demonstrate that the ice thicknesses and bed topographies we infer perform comparably to other inversions documented in the Ice Thickness Models Intercomparison eXperiment. Unlike other inversions, we do not require surface mass balance or glacier velocities, which can be inaccurate and difficult to obtain. Given the increasing availability of high-resolution surface elevation data, it may be possible to apply this method to glaciers worldwide to better constrain the ice thickness, bed topography and volume of glaciers globally.