Most efficient sections for area 6 m²

A most efficient section has the least wetted perimeter for a given area. Area $A=6$ m² in all cases.

(a) Triangular section

Best side slope is $z=1$ (45°): $A=y^2$, $P=2\sqrt2\,y$.

$$y=\sqrt6=2.449\ \text{m},\qquad \text{top width}=2y=4.899\ \text{m},\qquad P=2\sqrt2(2.449)=6.928\ \text{m}$$

(b) Rectangular section

$b=2y$, $A=2y^2$:

$$y=\sqrt{\frac62}=1.732\ \text{m},\qquad b=2y=3.464\ \text{m},\qquad P=4y=6.928\ \text{m}$$

(c) Trapezoidal section, side slope 2:1 (taken as 2H:1V, $z=2$)

$$A=y^2(2\sqrt{1+z^2}-z)=y^2(2\sqrt5-2)=2.4721\,y^2$$ $$y=\sqrt{\frac{6}{2.4721}}=1.558\ \text{m},\quad b=2y(\sqrt5-2)=0.736\ \text{m}$$ $$P=b+2y\sqrt5=0.736+2(1.558)(2.2361)=7.703\ \text{m}\quad(\text{also }P=2A/y)$$

Comparison

Answer: the triangular and the rectangular sections both have the least perimeter (6.93 m); the 2H:1V trapezoid has a larger perimeter (7.70 m).

Note: if "2:1" is read as 2V:1H ($z=0.5$), the trapezoid has $y=1.859$ m, $b=2.298$ m and $P=6.455$ m, which is the least of the three. The perimeter is smallest when the side slope is close to the half-hexagon value $z=0.577$.