A theoretical and experimental study on curvature sensing using a Brillouin optical time-domain analyzer based on the ring-core fiber (RCF) is reported. The Brillouin gain spectrum of the RCF is investigated, and the Brillouin frequency shift (BFS) dependence on temperature and strain is calibrated. We theoretically analyze the fiber bending-induced BFS and peak Brillouin gain variation for the RCF through a numerical simulation method, and the RCF is revealed to have a high curvature sensitivity. Distributed curvature sensing is successfully demonstrated, with the bending radius ranging from 0.5 to 1.5 cm, corresponding to a BFS variation from 32.90 to 7.81 MHz. The RCF takes advantage of great bending loss resistance, and the maximum macrobending loss at the extreme bending radius of 0.5 cm is less than 0.01 dB/turn. Besides, the peak Brillouin gain of the RCF is discovered to vary significantly in response to fiber bending, which is expected to be another parameter for distributed curvature determination. The results imply that the RCF is a promising candidate for highly sensitive distributed curvature measurement, especially in sharp bending circumstances.