In most metropolitan regions throughout the globe, urbanised land area is increasing to accommodate increasing population size. This article provides a simple yet powerful mathematical description of this urban expansion. Specifically, the following scaling relationship is proposed: land area (A) increases proportionally to population size (P) raised to a power (n) - i.e. A ∝ Pn. During 1950-2000, this relationship is found to hold well for US Census urban areas (UAs) with a greater than 10 per cent increase in population. Values for the parameter n vary among UAs, with a central tendency value of ∼ 2, suggesting that, on average, newcomers to urban areas occupy about twice the land area per capita of existing residents. If n were exactly equal to 2, then the parameter group P/√ A (called 'linear population density', or LPD) would be constant over time. LPD (units: people per metre) is the number of people in a metre-wide strip across an urban area. LPD is distinct from, and behaves somewhat differently than, population density. Distributions of LPD values among US UAs during 1950-2000 show surprisingly little variability over multidecade time-scales. For example, from 1950 to 2000, average population, land area and population density changed by more than a factor of 2, but average LPD changed less than 10 per cent. Few, if any, other attributes of urban form have remained so constant during this half-century time-period. International data corroborate the finding that LPD distributions are roughly constant over multidecadal time-scales. These results suggest an underlying pattern to how people arrange themselves within and among urban areas. For US UAs, rank-size rules similar to the generalised version of Zipf's rule hold for population, land area, LPD and population density. LPD is an important predictor of the emissions-to-inhalation relationship for motor vehicle emissions. Results presented here are important for theoretical, practical and empirical investigations of urban form and of how urban areas expand over time.