From geometry to sustainability: Optimal shapes of hip roof houses

TL;DR

This paper presents a mathematical framework for optimizing hip roof house shapes to minimize external surface area, aiding sustainable design through explicit formulas, case studies, and a supporting software tool.

Contribution

It introduces a rigorous optimization model for hip roof geometries, deriving explicit formulas and providing practical guidelines and software for sustainable architectural design.

Findings

Square footprints with balanced slenderness are most efficient.

Elongated or flattened shapes increase energy and material use.

The software aids designers in applying optimization methods.

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.