We examine the conditions under which a misfitting thin film on a substrate will twin and the pattern of the twin microstructure that will form. Using linear elasticity theory, we present exact analytical results for the energy released due to the formation of a periodic array of twinned domains as well as that for a single embedded twin domain. While the analysis is applicable to general misfit strain tensors, we specifically analyze the energetics of twinned domain formation for a tetragonal film on a cubic substrate. The analysis shows that for a periodic distribution of twins, the equilibrium width of the domains of the two variants are identical. Our results also show that the equilibrium periodicity of the microstructure scales inversely with film thickness for large film thickness and decays exponentially with increasing film thickness for small film thicknesses.