Electrical Impedance Tomography (EIT) problem consists of finding the electric conductivity inside a conductive body from simultaneous measurements of electric potential and current at its boundary. The EIT problem with partial boundary measurements corresponds to the case, where only certain portions of the boundary are available for measuring the potentials and currents. A deep connection exists between the EIT problem in continuum and the discrete inverse problem of recovering the conductances of resistors in a resistor network. The connection comes in the form of special finite volume discretizations on carefully chosen grids, known as optimal grids. The optimal grids allows us to use the existing theory of inverse problems for resistor networks to compute an approximation to the solution of the continuum EIT problem. This was done recently in the case of full boundary measurements. The main goal of our work is to generalize the existing results to the case of partial measurement settings. Two alternative approaches are presented: one is based on (quasi-) conformal mappings of a well-studied conductivity problem in a unit disk, and the other is based on a previously unstudied pyramidal graph topology. We establish the existence and uniqueness of the solutions of discrete problems for both settings. Then we use these discrete results to obtain numerical approximations to the solution of the continuum EIT problem with partial measurements.