We show that a drop of liquid a few hundred microns in diameter placed under a solid, elastic, thin film (∼10 μm thick) causes it to bulge by tens of microns. The deformed shape is governed by equilibrium of tensions exerted by the various interfaces and the solid film, a form of Neumann’s triangle. Unlike Young’s equation, which specifies the contact angles at the junction of two fluids and a (rigid) solid, and is fundamentally underdetermined, both tensions in the solid film can be determined here if the liquid–vapor surface tension is known independently. Tensions in the solid film have a contribution from elastic stretch and a constant residual component. The residual component, extracted by extrapolation to films of vanishing thickness and supported by analysis of the elastic deformation, is interpreted as the solid–fluid surface tension, demonstrating that compliant thin-film structures can be used to measure solid surface tensions.