Moderate earthquakes (Mw > 5) with moment tensors (MTs) dominated by a vertical compensated-linear-vector-dipole (vertical-CLVD) component are often generated by dip slip along a curved ring-fault system at active volcanoes. However, relating their MTs to ring-fault parameters has been proved difficult. The objective of this study is to find a robust way of estimating some ring-fault parameters based on their MT solutions obtained from long-period seismic records. We first model the MTs of idealized ring-faulting and show that MT components representing the vertical-CLVD and vertical strike-slip mechanisms are resolvable by the deviatoric MT inversion using long-period seismic waves, whereas a component representing the vertical dip-slip mechanism is indeterminate owing to a shallow source depth. We then propose a new method for estimating the arc angle and orientation of ring-faulting using the two resolvable MT components. For validation, we study a vertical-CLVD earthquake that occurred during the 2005 volcanic activity at the Sierra Negra caldera, Galápagos Islands. The resolvable MT components are stably determined with long-period seismic waves, and our estimation of the ring-fault parameters is consistent with the ring-fault geometry identified by previous geodetic studies and field surveys. We also estimate ring-fault parameters of two earthquakes that took place during the 2018 activity at the caldera, revealing significant differences between the two earthquakes in terms of slip direction and location. These results show the usefulness of our method for estimating ring-fault parameters, enabling us to examine the kinematics and structures below active volcanoes with ring faults that are distributed globally.