We study the domain geometry during spinodal decomposition of a 50:50 binary mixture in two dimensions. Extending arguments developed to treat nonconserved coarsening, we obtain approximate analytic results for the distribution of domain areas and perimeters during the dynamics. The main approximation is to regard the interfaces separating domains as moving independently. While this is true in the nonconserved case, it is not in the conserved one. Our results can therefore be considered as a "first-order" approximation for the distributions. In contrast to the celebrated Lifshitz-Slyozov-Wagner distribution of structures of the minority phase in the limit of very small concentration, the distribution of domain areas in the 50:50 case does not have a cutoff. Large structures (areas or perimeters) retain the morphology of a percolative or critical initial condition, for quenches from high temperatures or the critical point, respectively. The corresponding distributions are described by a cA-tau tail, where c and tau are exactly known. With increasing time, small structures tend to have a spherical shape with a smooth surface before evaporating by diffusion. In this regime, the number density of domains with area A scales as A1/2 , as in the Lifshitz-Slyozov-Wagner theory. The threshold between the small and large regimes is determined by the characteristic area A approximately t2/3. Finally, we study the relation between perimeters and areas and the distribution of boundary lengths, finding results that are consistent with the ones summarized above. We test our predictions with Monte Carlo simulations of the two-dimensional Ising model.