In this paper, we propose a mathematical model of a problem related to the determination of elasto-plastic properties of a deformable axisymmetric isotropic material using an experimentally given penetration diagram. The considered physical model is based on elasto-plastic deformation theory. The problem leads to an inverse coefficient problem for the non-linear system of equilibrium equations with an additional condition (experimentally measured penetration diagram). This inverse problem is reformulated as a minimization problem for a certain functional. By using Lagrange linear triangle elements, the finite element formulation is presented. The numerical algorithms for both direct and inverse problems are described. Several numerical examples of the considered problem solution are given to show the accuracy and reliability of the proposed method. The influence of measurement errors is examined in detail.