<p>A good understanding of cities is crucial to implement urban planning policies leading to social and economic sustainability and an efficient use of resources. Urban concentration has been associated with both positive and negative effects, echoing debates on compact cities, but we still lack robust empirical evidence on how intra-urban density systematically evolves with city size. We fill this gap by determining the functional form of the population density radial (“center-periphery”) profiles for a global sample of cities, and observing how it changes across the urban hierarchy. Our results uncover strong regularities in the decrease of these profiles with distance from the main center, and in its link with city size, as large cities are denser and more spread out than smaller ones. We actually find that cities can be seen, in terms of population density, as exponential cones that evolve homothetically with city population. This rather simple but universal geometric structure of cities provides a new spatial scaling law, which is an important step forward in understanding how cities work and grow. Some deviations can be observed, which mainly oppose dense cities in the developing world and sprawled cities in high-income countries, associated with high energy use per capita. This suggests that the urban lifestyle in the wealthiest countries has come at the price of negative impacts on environmental outcome. This research has a broad range of applications as it provides a powerful tool to compare cities of different sizes.</p>