There are two commonly discrete approximations for the inverse conductivity problem.Finite element models are heavily used in electrical impedance tomography research as they are easily adapted to bodies of irregular shapes. The other approximation is to use electrical resistor networks for which several uniqueness results and reconstruction algorithms are known for the inverse problem. In this thesis the link between finite element models and resistor networks is established. For the planar case we show how resistor networks associated with a triangular mesh have an isotropic embedding and we give conditions for the uniqueness of the embedding. Moreover, a layered finite element model parameterized by thevalues of conductivity on the interior nodes is constructed. Construction of the finite element mesh leads to a study of the triangulation survey problem. A constructive algorithm is given to determine the position of the nodes in the triangulation with a knowledge of one edge and the angles of the finite element mesh. Also we show that we need to satisfy the sine rule as aconsistency condition for every closed basic cycle that enclosing interior nodes and this is a complete set of independent constraints.