From geometry to sustainability: Optimal shapes of hip roof houses

Abstract

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 of the building envelope for 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. For each case, explicit formulas for the optimal dimensions are derived, offering architects and engineers practical guidelines for improving material efficiency, reducing construction costs, and enhancing energy performance. To illustrate the practical relevance of the theoretical results, case studies of real-world hip roof houses are presented, revealing both inefficiencies in common practice and near-optimal examples. Furthermore, a freely available software application has been developed to support designers in applying the optimization methods directly to architectural projects. The findings confirm that square-based footprints combined with balanced slenderness ratios yield the most efficient forms, while deviations toward elongated or flattened proportions significantly increase energy and material demands. This work demonstrates how mathematical modeling and architectural design can be integrated to support sustainable architecture, providing both theoretical insight and practical tools for shaping energy-efficient, cost-effective, and aesthetically coherent residential buildings.

Figures (15)

• Figure 1: Typical shape of the hip roof house with notation used in the paper. On the left side of the drawing we have a perspective view, on the right a view from above.
• Figure 2: Graph of the external surface $S(r,k)$ depending on the footprint aspect ratio $r$ and slenderness aspect ratio $k$, with parameters $V = 400 \; \text{m}^3$ and $\alpha = \pi/6 = 30^\circ$. The red dot indicates the location of the minimal surface.
• Figure 3: Contour plot of the external surface $S$ as a function of footprint aspect ratio $r$ and slenderness aspect ratio $k$. Here $\alpha=\pi/6=30^\circ$ and $V=400 \; m^2$. Red dot is the point corresponding to the global minimum equal to $S_{\min}=271,23\;m^2$, attained for $r=1$ and $k=0,58$.
• Figure 4: Plots of the optimal parameters $W_{\min}$, $L_{\min}$ and $H_{\min}$ as functions of $\alpha$. Here $V=400 \; m^3$.
• Figure 5: Graph of the ratio $S/S_{\min}$ depending on the footprint aspect ratio $r$ and slenderness aspect ratio $k$, with parameters $V = 400 \; \text{m}^3$ and $\alpha = \pi/6 = 30^\circ$. The red dot indicates the location of the minimum equal to 1.