In predator-prey interactions, the escape maneuvers of prey animals is a key determinant of their likelihood of survival. Therefore, these maneuvers and the neural circuits subserving them are under intense selection pressure. Our current understanding suggests that a number of escape parameters contribute to survival, including response latency, escape speed, and direction. However, existing studies present contradictory evidence about the impact of these parameters on escape success. Specifically, the value of rapid responses and fast speeds in producing successful escapes has been disputed, even while many animals have specialized circuits devoted to producing the shortest latency and fastest escapes. These contradictions obscure an understanding of the selection pressure on escape maneuvers and the functional benefit of specialized escape circuits. To clarify these issues, we have investigated the determinants of successful escape maneuvers by studying the responses of larval zebrafish to the ballistic attacks of a natural predator, the dragonfly nymph. We found that the strongest predictor of the outcome was the time needed for the nymph to reach the fish after the fish had initiated an escape. We show how this result is a consequence of the intersection of the volume containing all possible escape trajectories of the fish and the swept volume of the nymph's attack. By analyzing the interaction of these volumes, we estimated the survival benefit conferred to larval zebrafish by recruiting the Mauthner cell, the giant neuron in fish devoted to producing escapes. Our approach provides a new perspective on the selection pressure on movements during predation by presenting a general framework that unifies the influence of many escape parameters in shaping the motor volumes of predator and prey.