Flexible surgical instruments can flexibly adjust their posture with a high degree of freedom, which makes them highly suitable for performing surgical tasks in narrow workspaces. However, redundant degrees of freedom increase their kinematic difficulty, which may cause redundant solutions, complex calculations, and low speeds. In this paper, a flexible surgical instrument is presented. The structural characteristics of this flexible instrument were explored in terms of force balance, it was concluded that the instrument had a constant curvature during bending. Based on this, the kinematics and inverse kinematics were solved via the geometric and Newton iteration methods, respectively. Our experiments showed that the proposed method for solving flexible instrument kinematics had high precision, a unique solution, and high speed; the instrument can be well controlled to perform refined operations. The proposed geometric method for solving the flexible instrument kinematics avoided the calculation of the Jacobian matrix, making it fast and capable of meeting the master-slave control requirement for real-time surgery. Furthermore, the proposed kinematics solution method is not limited by the mechanical structure, so it can be used for flexible instruments owning to its constant curvature bending.