Four unrelated groups of large cruising vertebrates (tunas, whales, lamnid sharks and parvipelvian ichthyosaurs) evolved tuna-shaped (thunniform) body plans. Stringent physical constraints, imposed by the surrounding fluids, are probably responsible for this example of evolutionary convergence. Here I present a mathematical model of swimming kinematics and fluid mechanics that specifies and quantifies such constraints, and test the model with empirical data. The test shows quantitatively that morphology, kinematics, and physiology indeed covary tightly in large cruisers. The model enables calculations of optimal cruising speed from external measurements, and also predicts that wide caudal fin spans, typical of thunniform swimmers, are necessary for large cruisers. This finding is contrary to a popular yet rather teleological view that thunniform tails were selected for their high aspect ratios that increased propulsive efficiency. I also show by calculation that Stenopterygius, a Jurassic ichthyosaur, probably had optimal cruising speeds and basal metabolic rates similar to living tunas.