The mechanisms that determine bacterial shape are in many ways poorly understood. A prime example is the Lyme disease spirochete, Borrelia burgdorferi (B. burgdorferi), which mechanically couples its motility organelles, helical flagella, to its rod-shaped cell body, producing a striking flat-wave morphology. A mathematical model is developed here that accounts for the elastic coupling of the flagella to the cell cylinder and shows that the flat-wave morphology is in fact a natural consequence of the geometrical and material properties of the components. Observations of purified periplasmic flagella show two flagellar conformations. The mathematical model suggests that the larger waveform flagellum is the more relevant for determining the shape of B. burgdorferi. Optical trapping experiments were used to measure directly the mechanical properties of these spirochetes. These results imply relative stiffnesses of the two components, which confirm the predictions of the model and show that the morphology of B. burgdorferi is completely determined by the elastic properties of the flagella and cell body. This approach is applicable to a variety of other structures in which the shape of the composite system is markedly different from that of the individual components, such as coiled-coil domains in proteins and the eukaryotic axoneme.