Various aspects of the built environment have important effects on ecology. Providing suitable metrics for the built forms so as to quantify and model their internal relations and external ecological footprints, however, remains a challenge. Here we provide such metrics focusing on the spatial distribution of 11,418 buildings within the city of Geneva, Switzerland. The size distributions of areas, perimeters, and volumes of the buildings follow approximately power laws, whereas the heights of the buildings follow a bimodal (two-peak) distributions. Using the Gibbs–Shannon entropy formula, we calculated area, perimeter, volume, and height entropies for 16 neighbourhoods (zones) in Geneva and show that they have positive correlations (R2 = 0.43–0.84) with the average values of these parameters. Furthermore, the entropies of area, perimeter, and volume themselves are all positively correlated (R2 = 0.87–0.91). Deriving entropy from Helmholtz free energy, we interpret entropy as a measure of spreading or expansion and provide an analogy between the entropy increase during the expansion of a solid and the entropy increase with the expansion of the built-up area in Geneva. Compactness of cities is widely thought to affect their ecology. Here we use the density of buildings and transport infrastructure as a measure of compactness. The results show negative correlation (R2 = 0.39 0.54) between building density and the entropies of building area, perimeter, and volume. The calculated length-size distributions of the street network shows a negative correlations (R2 = 0.70–0.76) with the number of streets per unit area as well as with the total street length per unit area. The number of buildings as well as populations (number of people) show sub-linear relations with both the annual heat demand (MJ) and CO2 emissions (kg) for the 16 neighbourhoods. These relations imply that the heat demand and CO2 emissions grow at a slower rate than either the number of buildings or the population. More specifically, the relations can be interpreted so that 1% increase in the number of buildings or the population is associated with some 0.8–0.9% increase in heat demand and CO2 emissions. Thus, in terms of number of buildings and populations, large neighbourhoods have proportionally less ecological footprints than smaller neighbourhoods.