The ocean supplies energy for tropical cyclones (TCs) and slows their winds through surface friction, which exerts a force on the ocean termed wind stress. The drag coefficient (C_d) is the key parameter that converts wind speed to wind stress and is currently estimated in forecast models from incomplete data collected in low-to-moderate ocean winds. Here, we use measurements from 11 Atlantic hurricanes to quantify C_d in winds up to 44 meters per second and surface waves up to 14 meters. It is found that C_d levels off, as wind speed surpasses 30 meters per second but does not decrease appreciably as suggested by previous indirect methods. Interaction of the wind and wave fields causes C_d to be 20 ± 2% higher on the motion-left side of a storm, where wind and waves are misaligned, than on the right. These results quantify directly a fundamental TC air-sea interaction parameter and demonstrate the importance of distinct TC quadrant-specific wind-wave interactions.