Designing sustainable barn-type houses: Optimal shapes for minimal envelope and energy use

TL;DR

This paper presents a mathematical framework for optimizing barn-type house geometries to minimize envelope area and energy use, providing explicit formulas and software tools for practical design improvements.

Contribution

It introduces a novel optimization method for barn house shapes, including explicit solutions and software tools, to enhance energy efficiency and reduce costs.

Findings

Optimal shapes significantly reduce surface area and energy use.

Design deviations from optimal compactness vary among existing houses.

The methodology enables cost-effective, energy-efficient architectural designs.

Abstract

Barn-type houses have become one of the most popular single-family housing typologies in Poland and across Europe due to their simplicity, functionality, and potential for energy efficiency. Despite their widespread use, systematic methods for optimizing their geometry in terms of envelope area and energy performance remain limited. This paper develops a rigorous mathematical framework for determining the optimal proportions of barn-type houses with respect to minimizing the external surface area while satisfying constraints of either fixed volume or fixed floor area. Closed-form solutions for the optimal width, length, and height are derived as explicit functions of the roof slope, together with formulas for the minimal achievable surface. A recently introduced dimensionless compactness measure is also calculated, allowing quantitative assessment of how far a given design deviates from the theoretical optimum. The methodology is applied to case studies of three existing houses, showing that while some designs deviate significantly from optimal compactness, others already closely approximate it. The results confirm that theoretical optimization can lead to meaningful reductions in construction costs and energy demand. To support practical implementation, two original freely available software tools were developed, enabling architects and engineers to perform optimization analyses.