The highly efficient coupling of light from conventional optical components to optical mode volumes lies in the heart of chip-based micro-devices, which is determined by the mode-matching between propagation constants of fiber taper and the whispering-gallery-mode (WGM) of the resonator. Optical gyroscopes, typically realized as fiber-optic gyroscopes and ring-laser gyroscopes, have been the mainstay in diverse applications such as positioning and inertial sensing. Here, the mode-matching is theoretically analyzed and experimentally verified. We observe the Sagnac effect in a millimeter-scale wedged resonator gyroscope, which has attracted considerable attention and has been rapidly promoted in recent years. We demonstrate a bidirectional pump and probe scheme, which directly measures the frequency beat caused by the Sagnac effect. We establish the linear response between the detected beat frequency and the rotation velocity. The clockwise and counterclockwise rotation can also be distinguished according to the value of the frequency beat. The experimental results verify the feasibility of developing the gyroscope in a WGM resonator system and pave the way for future development.