The classical definition of population density used in social science disciplines as a measure of density or physical crowding has severe shortcomings since actual spatial orientation is disregarded. A formula is proposed to measure physical crowding or population density which corrects the deficiency of the conventional formulation. As a generalization of the Euclidean distance formula, the proposed index measures relative population density as a function of spatial orientation and size of geometric area. Three mathematical properties of the index stated as conjectures have been supported by Monte Carlo simulations. The formula is evaluated and justified on three key scientific criteria: parsimony, mathematical soundness and empirical verifiability. Crowding is an obvious problem in the small compartmentalized workspaces of U.S. Navy nuclear submarines. The measurement of physical crowding by use of the proposed formula in Concepts of Operations Experiments conducted at the Newport Laboratory of the Naval Underwater Systems Center is reported and discussed.