Minimum roof length at fixed wall height and cross-sectional area ($2\tan 2\theta =w/h,w^4=16A(A-wh)$)

For $A,h>0$, a rectangular cross section of width $w$ and height $h$, capped by a symmetric triangular roof at angle $0<\theta <\pi /2$, has area $A=wh+w^2\tan \theta /4$. Its total sloping roof length is $w\sec \theta$. A cost $2\alpha h+\beta w\sec \theta$ with $\beta >0$ has the same minimizing shape as this roof length, since the fixed wall term is constant.

The Lagrange multiplier equations for $w\sec \theta$ under the area constraint give $w(1-{\tan }^2\theta )=4h\tan \theta$. This is the first displayed condition; it also implies $\theta <\pi /4$. Eliminating $\tan \theta =4(A-wh)/w^2$ gives the second condition.

For a global check, the squared roof length is $Q(w)=w^2+16(A-wh{)}^2/w^2$ for $0<w<A/h$. Its derivative has the sign of $w^4+16Ahw-16A^2$, a strictly increasing expression with one root in that interval. Thus this root is the unique minimum. The shape is independent of the positive roof-cost coefficient and the constant wall cost. If $\beta =0$, every feasible shape has the same cost and these conditions are no longer necessary.