Buildings generate carbon in two ways: operational emissions from heating, cooling, and lighting during use, and embodied emissions from producing and constructing their materials. Compact building forms are conventionally assumed to be environmentally superior because a small envelope-to-floor ratio cuts transmission losses, and the embodied side is often assumed to follow suit. However, compactness also reshapes structural material demand, daylight availability, and, at the urban-block scale, density-related morphology factors, so the two carbon components need not move together, and their sum could possess an interior optimum rather than improving monotonically with compactness. The literature has established component-wise relations only within limited ranges (a linear compactness–embodied relation for simple prismatic blocks; monotone improvements of both components over sampled compactness ranges; non-monotonicity along other geometry axes such as storey number and tallness), while the shape of the compactness–whole-life-carbon curve itself, and at which scales it holds, has not been characterized. The problem is to determine whether whole-life-cycle carbon varies non-monotonically with compactness — and if so where the interior optimum lies and which mechanisms (envelope, structure, lighting, block morphology) control it — across scales from the individual building to the urban block.

Background

Buildings account for roughly a third of global energy use and about 37% of energy-related carbon emissions, split into two life-cycle components. Operational carbon is emitted while the building is used — for heating, cooling, ventilation, lighting, and appliances — while embodied carbon is emitted before use, in extracting, manufacturing, transporting, and assembling construction materials. Life cycle assessment (LCA) quantifies both, usually normalized to a functional unit such as kilograms of CO2_2-equivalent per square meter of gross floor area (GFA, the total floor area within the building envelope) per year, with explicit system boundaries defining which life-cycle stages are counted.

A building's compactness describes how much envelope surface encloses a given amount of floor space; it is commonly measured by a shape coefficient such as envelope surface divided by GFA (smaller = more compact), or its inverse. Compact forms lose less heat through the envelope, and because envelope elements (walls, roofs, insulation, cladding) carry embodied carbon too, compactness plausibly reduces both components at once. Three strands of prior work shape the open question. First, Lotteau, Loubet and Sonnemann (2017) built a functional-element model (decomposing a concrete residential building into twelve elements whose material quantities follow from geometry) for French collective housing and found that for simple prismatic block geometries the compactness ratio CF2_2 = S/GFA correlates linearly with embodied energy and embodied carbon — a monotone relation over their sampled range — with operational energy outside the model; the work was framed as a simplified assessment tool tied to urban planners' design levers, and the associated doctoral research program extended toward assessing passive-heating and passive-summer-comfort potential at the neighborhood scale, so the generalization of the compactness–embodied relation to wider geometries and its combination with operational assessment (heating, cooling, lighting) remained open directions rather than results of that study. Second, Gauch, Dunant, Hawkins and Cabrera Serrenho (2023) performed a Sobol sensitivity study (a variance-decomposition method ranking each design variable's contribution to output variance) of multi-storey residential and office buildings, in which increasing compactness improved heating and cooling loads, embodied carbon, and construction cost in approximately hyperbolic, monotone fashion over the sampled range — yet the same study found a clear interior optimum in the number of storeys (mean embodied carbon minimized around 4–6 storeys), driven by the competing embodied shares of the roof versus columns and lateral stability systems. Third, Nazari, Dixit, Yan and Aryal (2024) confirmed through a literature review that studies simultaneously considering operational and embodied impacts in building shape optimization were lacking, and showed in two parametric case studies (differing only in insulation material) that the optimal building shape shifts once embodied energy is included alongside operational energy; Kamazani, Dixit and Shanbhag (2025) introduced an 'embodied energy factor' quantifying the embodied energy spent to save one unit of operational energy, noting that optimizing operational energy alone can inadvertently raise embodied energy.

At larger scales the picture diverges further. Pomponi, Saint, Arehart, Gharavi and D'Amico (2021) decoupled urban density from tallness and found, for real neighborhoods, that taller environments substantially increase life-cycle greenhouse-gas emissions (+154%) while densifying without adding height reduces them — an interior-optimum-like structure along the vertical dimension of urban form. Wang, Fei, Song and Li (2025) computed both embodied and operational carbon for 30 typical residential blocks in Beijing and found the two components associating differently with morphology metrics (enclosed blocks lowest, intensive highest; plot ratio and mean building height negatively correlated with both components, building density positively). Meanwhile, recent shape-focused studies remain single-component: office-shape LCA restricted to the operational stage, geometric compactness indicators for early-stage housing design, and closed-form envelope-area minimization all treat compactness as monotonically beneficial or address only one carbon component. Empirically, Torres-Quezada et al. (2025) showed in a warm-humid climate (54 Ecuadorian houses over four decades) that rising embodied energy did not reduce operational energy — evidence that the embodied–operational tradeoff is region- and climate-dependent rather than universal. Taken together, the individual pieces leave the central relation uncharacterized: nobody has traced whole-life-cycle carbon as a function of compactness across a wide range, established whether the curve is non-monotonic with an interior optimum, or reconciled the apparently scale-dependent behavior between single buildings and urban blocks.

Problem Statement

Determine how whole-life-cycle carbon — the sum of embodied and operational carbon per functional unit, with system boundaries and lifespan stated — varies as a function of building-form compactness (a stated shape coefficient such as envelope surface per unit floor area, varied over the widest range realizable for the building type): in particular, establish whether the relation is non-monotonic, whether it possesses an interior optimum strictly inside the feasible compactness range (as opposed to improving monotonically toward the most compact form), and identify the mechanisms (envelope-driven transmission losses and envelope material quantities, structural-material shares, daylight and artificial-lighting demand, and, at the block scale, density/height/mutual-shading morphology factors) that control the sign and magnitude of each component's dependence. The characterization is required at each of the three scales — individual building, building ensemble, and urban block — and comprises distinct, independently settled sub-claims: whether the relation is non-monotonic; whether it attains an interior optimum strictly inside the feasible compactness range, and where; and which mechanisms control the sign and magnitude of each component — with the compactness-metric definition used and the conditions on which its conclusions depend (climate, structural system, material set, energy-system carbon intensity) stated, since prior work shows that metric choice and correlated geometric variables (storey count, plan depth, window-to-wall ratio) can shift or reverse the relationship.

References

  1. Lotteau, M., Loubet, P., Sonnemann, G. (2017). An analysis to understand how the shape of a concrete residential building influences its embodied energy and embodied carbon. Energy and Buildings 154:1–11. DOI 10.1016/j.enbuild.2017.08.048. https://doi.org/10.1016/j.enbuild.2017.08.048
  2. Nazari, F., Dixit, M., Yan, W., Aryal, A. (2024). Building shape optimization based on interconnected embodied and operational energy and carbon impacts. Energy and Buildings 325:114933. DOI 10.1016/j.enbuild.2024.114933. https://doi.org/10.1016/j.enbuild.2024.114933
  3. Kamazani, M. A., Dixit, M. K., Shanbhag, S. S. (2025). Optimizing interconnected embodied and operational energy of buildings: An embodied energy factor approach. Building and Environment 276:112902. DOI 10.1016/j.buildenv.2025.112902. https://doi.org/10.1016/j.buildenv.2025.112902
  4. Wang, J., Fei, Z., Song, L., Li, R. (2025). Impact of urban morphology on dwellings carbon emissions at the district scale. Proceedings of the Institution of Civil Engineers – Engineering Sustainability 178(3):218–234. DOI 10.1680/jensu.23.00086. https://doi.org/10.1680/jensu.23.00086
  5. Gauch, H. L., Dunant, C. F., Hawkins, W., Cabrera Serrenho, A. (2023). What really matters in multi-storey building design? A simultaneous sensitivity study of embodied carbon, construction cost, and operational energy. Applied Energy 333:120585. DOI 10.1016/j.apenergy.2022.120585. https://doi.org/10.1016/j.apenergy.2022.120585
  6. Pomponi, F., Saint, R., Arehart, J. H., Gharavi, N., D'Amico, B. (2021). Decoupling density from tallness in analysing the life cycle greenhouse gas emissions of cities. npj Urban Sustainability 1(1):33. DOI 10.1038/s42949-021-00034-w. https://doi.org/10.1038/s42949-021-00034-w
  7. Torres-Quezada, J., Isalgue-Buxeda, A., Pages-Ramon, A., Lopez-Besora, J., Delgado-Bohorquez, A. (2025). Does higher embodied energy lead to lower operational energy? Insights from residential buildings in Ecuador's Costa Region. Journal of Building Engineering 113:113870. DOI 10.1016/j.jobe.2025.113870. https://doi.org/10.1016/j.jobe.2025.113870
  8. Shokouhi, S., Weidlich, I. (2025). An LCA Study of Various Office Building Shapes Focusing on Operational Energy—A Case of Hamburg. Sustainability 17(4):1659. DOI 10.3390/su17041659. https://doi.org/10.3390/su17041659
  9. Koźniewski, E. (2025). Assessment of Building Compactness at Initial Design Stage of Single-Family Houses. Energies 18(13):3569. DOI 10.3390/en18133569. https://doi.org/10.3390/en18133569
  10. Rokita-Magdziarz, E., Gronostajska, B., Magdziarz, M. (2026). From geometry to sustainability: optimal shapes of hip roof houses. Frontiers in Energy Research 14:1856004. DOI 10.3389/fenrg.2026.1856004. https://doi.org/10.3389/fenrg.2026.1856004