Head and neck surgeries require skilled surgeons to avoid facial nerves and blood vessels. Using straight surgical instruments to create a deep and sealed operating environment is difficult. Continuum robots are particularly effective in head and neck surgery due to their extensive range of motion, adaptability, and ability to navigate confined anatomical spaces. This paper presents a new approach to examine the flexibility of the continuum robot in head and neck surgery, motivated by the advantages of continuum robots. This method uses a discretized kinematics model with a constant curvature and a piecewise assumption-fitting-approximation (PFA) method to find closed solutions for the continuum robot’s problem. We introduced the concept of spatial solid angle to assess the dexterity of the continuum robot, utilizing the inverse solution of the target point position as a reference. An intuitive dexterity index and dexterity area were proposed. The length of the continuum segment was optimized by simulation, and the efficacy of the approach was then validated through experiments. Experiments demonstrate that the piecewise assumption can cause the dual-section continuum robot to deform as expected. In comparison to previous similar tasks, the robot has exceptional position accuracy, with a position error of only 1.72% of the overall length. Additionally, it exhibits excellent real-time performance, with a maximum time consumption of 3.9 ms.