Ultrasound computed tomography (USCT) is an emerging medical imaging modality that holds great promise for breast cancer diagnosis. Full-waveform inversion (FWI)-based image reconstruction methods for USCT can produce high spatial resolution and accurate images of the acoustic properties of soft tissues. A common USCT design employs a circular ring-array comprised of elevation-focused ultrasonic transducers. Volumetric imaging can be achieved by translating the ring-array orthogonally to the imaging plane. Slice-by-slice two-dimensional (2D) reconstruction methods have been implemented to form a composite three-dimensional (3D) volumes by stacking together reconstructed cross-sectional images at each ring-array position. However, this 2D approach does not account for the 3D wave propagation physics and the focusing properties of the transducers, and can result in out-of-plane scattering-based artifacts and inaccuracies. To overcome this, a new 3D time-domain FWI method is proposed for ring-array-based USCT that concurrently utilizes measurement data acquired from multiple positions of the ring-array. A virtual imaging study of ring-array-based USCT that employs a realistic 3D numerical breast phantom was conducted to assess the impact of the number of ring-array measurements on image quality.