The ability of proteins to sense membrane curvature is essential to cellular function. All known sensing mechanisms rely on protein domains with specific structural features such as wedge-like amphipathic helices and crescent-shaped BAR domains. Yet many proteins that contain these domains also contain large intrinsically disordered regions. Here we report that disordered domains are themselves potent sensors of membrane curvature. Comparison of Monte Carlo simulations with in vitro and live-cell measurements demonstrates that the polymer-like behavior of disordered domains found in endocytic proteins drives them to partition preferentially to convex membrane surfaces, which place fewer geometric constraints on their conformational entropy. Further, proteins containing both structured curvature sensors and disordered regions are more than twice as curvature sensitive as their respective structured domains alone. These findings demonstrate an entropic mechanism of curvature sensing that is independent of protein structure and illustrate how structured and disordered domains can synergistically enhance curvature sensitivity.