For systems of fused-hard-sphere molecules, we relate cavity distributions to free-volume distributions and derive new statistical geometry relations between thermodynamic and cavity properties. From these results, we formulate two different approaches for calculating the pressure of systems of two-dimensional homonuclear hard dumbbells in terms of the geometry of the free volume of a molecule observed over the course of a simulation. We find a simple linear relationship to exist between the compressibility and the average of the ratio of the surface area and volume of the angle-averaged free volumes of the dumbbells. We find excellent agreement between the pressure as calculated via both approaches and the pressure as calculated from simulations.