A fundamental evolution equation is developed to describe the distribution of areas of capture zones (CZs) associated with islands formed by homogeneous nucleation and growth during submonolayer deposition on perfect flat surfaces. This equation involves various quantities which characterize subtle spatial aspects of the nucleation process. These quantities in turn depend on the complex stochastic geometry of the CZ tessellation of the surface, and their detailed form determines the CZ area distribution (CZD) including its asymptotic features. For small CZ areas, behavior of the CZD reflects the critical island size, i. For large CZ areas, it may reflect the probability for nucleation near such large CZs. Predictions are compared with kinetic Monte Carlo simulation data for models with two-dimensional compact islands with i = 1 (irreversible island formation by diffusing adatom pairs) and i = 0 (adatoms spontaneously convert to stable nuclei, e.g., by exchange with the substrate).