Atta leaf-cutter ants are the prime herbivore in the Neotropics: differently sized foragers harvest plant material to grow a fungus as a crop. Efficient foraging involves complex interactions between worker size, task preferences and plant-fungus suitability; it is, however, ultimately constrained by the ability of differently sized workers to generate forces large enough to cut vegetation. In order to quantify this ability, we measured bite forces of Atta vollenweideri leaf-cutter ants spanning more than one order of magnitude in body mass. Maximum bite force scaled almost in direct proportion to mass; the largest workers generated peak bite forces 2.5 times higher than expected from isometry. This remarkable positive allometry can be explained via a biomechanical model that links bite forces with substantial size-specific changes in the morphology of the musculoskeletal bite apparatus. In addition to these morphological changes, we show that bite forces of smaller ants peak at larger mandibular opening angles, suggesting a size-dependent physiological adaptation, probably reflecting the need to cut leaves with a thickness that corresponds to a larger fraction of the maximum possible gape. Via direct comparison of maximum bite forces with leaf mechanical properties, we demonstrate (i) that bite forces in leaf-cutter ants need to be exceptionally large compared with body mass to enable them to cut leaves; and (ii), that the positive allometry enables colonies to forage on a wider range of plant species without the need for extreme investment in even larger workers. Our results thus provide strong quantitative arguments for the adaptive value of a positively allometric bite force.