Dispersed domains in two-phase systems often exhibit complex and intriguing morphologies. For many of these systems, it is possible to predict the shape of such a domain through an evaluation of the free energy. In this paper, we present a numerical technique to calculate domain shapes through a variational approach. This is a significant extension to previous work which has primarily involved examining free energies of specific classes of shapes. The solution diagram is presented in the neighborhood of the first four bifurcations from a circle and the affect of domain size is examined. We find the only stable domain shapes are circular or-bilobed. More importantly, we find that the transition between these two shapes is discontinuous, contrary to the findings of previous:shape studies. Examining the free energy functional in the neighborhood of stable solutions, we find that, in general, domain shape calculations are best applied to small nonelongated domains. Finally, we present graphical differentiation, a numerical technique we developed to enable solution of the problem.