The relationship between the two common measures of tropical cyclone intensity, the central pressure deficit and the peak near-surface wind speed, is a long-standing problem in tropical meteorology that has been approximated empirically yet lacks physical understanding. Here we provide theoretical grounding for this relationship. We first demonstrate that the central pressure deficit is highly predictable from the low-level wind field via gradient wind balance. We then show that this relationship reduces to a dependence on two velocity scales: the maximum azimuthal-mean azimuthal wind speed and half the product of the Coriolis parameter and outer storm size. This simple theory is found to hold across a hierarchy of models spanning reduced-complexity and Earth-like global simulations and observations. Thus, the central pressure deficit is an intensity measure that combines maximum wind speed, storm size, and background rotation rate. This work has significant implications for both fundamental understanding and risk analysis, including why the central pressure better explains historical economic damages than does maximum wind speed.