A gyroscope using two pairs of degenerate modes for wide-range sensing is proposed. We focused on a coaxial resonator composed of a cylinder and a column. To evaluate the characteristics of the proposed gyroscope, we analyzed its frequency response and rotational response both experimentally and numerically. From the frequency response under rotation, there was a difference between the shifts of resonance frequencies of two detecting modes. We also found that the lower-frequency mode had higher sensitivity and the higher-frequency mode had a wider linear region from the rotation response of each mode. These results suggest that high sensitivity for low-speed rotation and a wide linear region for high-speed sensing can be attained simultaneously by a single resonator with the help of two different pairs of degenerate modes.