Optimizing barn-type house geometry for minimum envelope area and first-order transmission heat-loss reduction

Barn-type houses are used in contemporary single-family residential architecture because of their simple gable-roof geometry, functional clarity, and compatibility with compact building forms. This paper presents the derivation of closed-form solutions for optimizing barn-type house geometry by minimizing the above-grade external surface area under fixed-volume and fixed-floor-area constraints. Explicit formulas are obtained for the optimal width, length, wall height, and minimum envelope area as functions of the roof slope angle. The study also applies an existing dimensionless compactness concept to the specific case of barn-type geometries, using it to quantify deviations from the theoretical envelope-area minimum. The proposed formulas and compactness assessment are demonstrated through selected illustrative case studies of existing barn-type houses. The results indicate that deviations from the constrained optimum can lead to measurable envelope-area penalties. Under the stated simplified assumptions, reductions in envelope area may be used to derive approximate, first-order estimates of envelope-related cost implications and transmission heat-loss reduction. The proposed method should therefore be understood as an early-stage geometric decision-support tool, rather than as a complete assessment of building energy demand, construction cost, life-cycle performance, or sustainability.