Vehicle passing angles are critical metrics for evaluating the geometric passability of vehicles. The accurate measurement of these angles is essential for route planning in complex terrain and in guiding the production of specialized vehicles. However, the current measurement methods cannot meet the requirements of efficiency, convenience and robustness. This paper presents a novel measurement method by building and measuring the point cloud of a vehicle chassis. Based on this method, a novel measurement system is designed and its effectiveness is verified. In the system, a wheeled robot acquires and processes data after passing underneath the vehicle. Then, we introduce a new approach to reduce the main sources of error when building point clouds beneath the vehicle, achieved by modifying the extraction algorithm and the proportion of different feature points in each frame. Additionally, we present a fast geometric calculation algorithm for calculating the passing angles. The simulation experiment results demonstrate deviations of 0.06252%, 0.01575%, and 0.003987% when comparing the calculated angles to those of the simulated vehicle. The experimental results show that the method and system are effective at acquiring the point cloud of the vehicle and calculating the parameters of passing angles with good data consistency, exhibiting variances of 0.12407, 0.12407, and 0.69804.