We apply the Lagrangian technique to compute and analyze the magnetization and the magnetic torque on soft-magnetic shape-dominant structures. The main advantages are the considerable reduction of the computation time, constant computation times across the saturation limits, and a closed-form description of the magnetization in both the linear and the saturation region. The analysis of the Lagrange multiplier allows, for the first time, the analytic prediction of the evolution of the magnetic torque as a function of the direction and magnitude of the applied field. In addition, it allows one to quantify the saturation state of a body, which is then used to predict the torque on assembled structures where the classic method fails. The results are verified by experiments and finite element analysis.