The capacity of organisms to adapt to vacant niches or changing environments is limited by physical constraints on morphological evolution. Substantial progress has been made in identifying how these constraints shape the form and function of producers (plants), but our understanding of evolutionary limits in consumers (animals) remains highly limited, in part because the requisite data have not been available at sufficient scale. Using morphometric measurements for all birds, we demonstrate that observed variation is highly restricted, both for beak shape and overall body shape, to triangular regions of morphospace with clearly defined boundaries and vertices. By combining morphometric data with new information on physical functions of measured traits, we provide evidence that the extent of avian morphospace is constrained by biophysical trade-offs between three functional objectives (strength, reach and engulfment capacity) that characterize resource acquisition and processing by the beak, and three locomotory modalities (aerial, aquatic and terrestrial) that characterize avian lifestyles. Our results suggest that over avian evolutionary history, trajectories of morphological change trend towards the vertices, with birds evolving from a core of biophysical generalists to biophysical specialists, associated with faster macroevolutionary turnover of lineages at the periphery or morphospace. Our analyses reveal that the structure of avian morphological diversity follows relatively simple rules defined by biophysical constraints and trade-offs, shedding light on the process shaping modern animal diversity and responses to environmental change.