Leaf dry mass per unit leaf area (LMA) is a central trait in ecology, but its anatomical and compositional basis has been unclear. An explicit mathematical and physical framework for quantifying the cell and tissue determinants of LMA will enable tests of their influence on species, communities and ecosystems. We present an approach to explaining LMA from the numbers, dimensions and mass densities of leaf cells and tissues, which provided unprecedented explanatory power for 11 broadleaved woody angiosperm species diverse in LMA (33-262 g m^-2 ; R²  = 0.94; P < 0.001). Across these diverse species, and in a larger comparison of evergreen vs. deciduous angiosperms, high LMA resulted principally from larger cell sizes, greater major vein allocation, greater numbers of mesophyll cell layers and higher cell mass densities. This explicit approach enables relating leaf anatomy and composition to a wide range of processes in physiological, evolutionary, community and macroecology.