The shape of motile cells is determined by many dynamic processes spanning several orders of magnitude in space and time, from local polymerization of actin monomers at subsecond timescales to global, cell-scale geometry that may persist for hours. Understanding the mechanism of shape determination in cells has proved to be extremely challenging due to the numerous components involved and the complexity of their interactions. Here we harness the natural phenotypic variability in a large population of motile epithelial keratocytes from fish (Hypsophrys nicaraguensis) to reveal mechanisms of shape determination. We find that the cells inhabit a low-dimensional, highly correlated spectrum of possible functional states. We further show that a model of actin network treadmilling in an inextensible membrane bag can quantitatively recapitulate this spectrum and predict both cell shape and speed. Our model provides a simple biochemical and biophysical basis for the observed morphology and behaviour of motile cells.