Presented in this paper is a numerical method for calculating the intersection points between Z-map vectors and the tool swept surface for circularly moving filleted-end mills. In numerically controlled(NC) machining simulation for large moulds and dies, a workpiece is frequently approximated as a set of z-axis aligned vectors, called Z-map vectors, and then the machining processes can be simulated through updating the Z-map with the intersection points. Circular motions are typically used for machining the free-form surfaces. For fast computation, we express each of intersection points with a single-variable non-linear equation and calculate the candidate interval in which the unique solution exists. Then, we prove the existence of a solution and its uniqueness in this candidate interval. Based on these properties, we can effectively apply numerical methods to finally calculate the solution of the nonlinear equation within a given precision. Experimental results are given for the case of a TV monitor and the hood of a car.