From geometry to sustainability: optimal shapes of hip roof houses

In this paper, we develop a rigorous mathematical framework for the optimization of hip roof house geometry, with the primary goal of minimizing the external surface area of the building envelope under a given set of design constraints. Five optimization scenarios are systematically analyzed: fixed volume, fixed footprint ratio, fixed slenderness ratio, fixed floor area, and constrained height within a prescribed range. For each case, explicit closed-form formulas for the optimal dimensions are derived, providing architects and engineers with practical guidelines for shaping more compact, material-efficient, and energy-conscious residential buildings. To demonstrate the practical relevance of the framework, the derived formulas are applied to three real-world hip roof houses. The case studies reveal that the proposed optimization can reduce the envelope area by up to hundred square meters, which corresponds to estimated construction-cost savings of approximately several thousand EUR and annual transmission-heat-loss reductions of more than one thousand k W h / y e a r in the most unfavorable analyzed case. Even under more restrictive design assumptions, the achievable savings remain significant, confirming the direct economic and energetic value of compactness-oriented design. The obtained results are particularly useful at the early stage of architectural design, when the main geometric parameters of the building, such as footprint proportions, height, and roof slope, can still be adjusted with relatively low effort and cost. At this conceptual stage, the proposed formulas can support rational decision-making by helping designers identify inefficient massing solutions before detailed functional, structural, and technological choices are introduced. To facilitate implementation in practice, a freely available software application has been developed, enabling designers and students to directly apply the optimization procedures in architectural workflows. The presented results demonstrate that mathematical modeling can serve as an effective decision-support tool in sustainable housing design, helping to identify inefficient geometric configurations, quantify their consequences, and indicate design modifications that improve both economic performance and thermal efficiency. The study thus confirms that the integration of geometry, optimization, and sustainability can provide practical and immediately applicable guidance for the design of energy-efficient, cost-effective, and environmentally responsible hip roof houses.