Correlation of sharp contact problems is investigated with the focus on rigid-plastic contact behavior pertinent to engineering metals and alloys. The aim is to determine relations between the contact hardness and constitutive parameters suitable for material characterization. This is performed by using a solution approach where the transition zone between elastoplastic and rigid-plastic contact behavior is analyzed, especially as regards the size of the plastic zone. This approach is applied to three different cases: (1) sharp indentation of von Mises plastic materials; (2) sharp indentation of Drucker-Prager plastic materials; (3) sharp scratching of von Mises plastic materials. Established finite element simulations are used in order to verify the analysis of these three cases. In addition, based on the suggested approach, new results concerning the constitutive parameter dependence of the relative contact area, pertinent to case (2) above, are presented and compared with finite element simulations. All of the results are valid for metals and alloys but also for elastic-plastic crystalline materials where the contact deformation regime is close to the rigid-plastic one.