This paper reports the results of a theoretical and experimental study of how the initial circulation of ring-shaped Bose-Einstein condensates (BECs) can be probed by time-of-flight (TOF) images. We have studied the dynamics of a BEC after release from a toroidal trap potential by solving the three-dimensional Gross-Pitaevskii (GP) equation. The trap and condensate characteristics matched those of a recent experiment. The circulation, experimentally imparted to the condensate by stirring, was simulated by imprinting a linear azimuthal phase on the initial condensate wave function. The simulated TOF images were in good agreement with the experimental data. We find that upon release the dynamics of the ring-shaped condensate proceeds in two distinct phases. First, the condensate expands rapidly inward, filling in the initial hole until it reaches a minimum radius that depends on the initial circulation. In the second phase, the density at the inner radius increases to a maximum after which the hole radius begins slowly to expand. During this second phase a series of concentric rings appears due to the interference of ingoing and outgoing matter waves from the inner radius. The results of the GP equation predict that the hole area is a quadratic function of the initial circulation when the condensate is released directly from the trap in which it was stirred and is a linear function of the circulation if the trap is relaxed before release. These scalings matched the data. Thus hole size after TOF can be used as a reliable probe of initial condensate circulation. This connection between circulation and hole size after TOF will facilitate future studies of atomtronic systems that are implemented in ultracold quantum gases.