Contact is a phenomenon faced in every day life, which is actually a complex problem to tackle for engineers. Most of the times, may be impossible to get analytic or exact results for the interaction of bodies in contact. In this thesis work, solution of the frictionless contact of an elastic body, touching to a rigid planar surface for two-dimensional elasticity / namely plane stress, plane strain and axi-symmetric formulations is aimed. The problem is solved numerically, with Finite Element Method, and an Object Oriented computer program in C++ for this purpose is written, and the results are verified with some basic analytic solutions and ABAQUS package program. It is not aimed in this thesis work to give a new solution in the area of solution of contact problems, but instead, it is aimed to form a strong basis, and computational library, which is extendible for further development of the subject to include friction, plasticity, and different material modeling in this advanced field of mechanics.