An all-fiber Mach-Zehnder interferometer (MZI) using ring core few-mode fiber (RC-FMF) for curvature sensing is proposed and experimentally demonstrated. The MZI was fabricated by splicing a segment of RC-FMF between two pieces of single-mode fiber (SMF). With the benefit of a RC of the central axis of the RC-FMF, the sensor is more sensitive to curvature compared to other fiber sensors based on ordinary SMF or FMF. Curvature measurement can be achieved by monitoring the wavelength shift of interference dips. Experimental results have shown that the sensitivity of curvature sensing can reach up to -4.370nm/m-1, within the range of 1.199-1.549m-1. Also, the temperature sensing characteristics of the sensor are measured, and the maximum temperature sensitivity is 57.6 pm/°C, ranging from 25°C to 45°C. The proposed MZI sensor has excellent potential for curvature measurement of building structural health monitoring, bridge engineering, and more.