Part A: Trapezoidal section
← Equation (1)
Differentiate partially with respect to d
From Equation (1)
From x = b + 2d tan θ
Therefore, the most efficient trapezoidal section has its top width equal to the sum of the sides, which is a proportion for a half-hexagon.
Again from Equation (1)
This shows that the best of all efficient trapezoidal section is the regular half-hexagon (all sides are equal).
Part B: Triangular section
From the equation of cross-sectional area
Construct the angle θ/2 and find the hypotenuse by Pythagorean theorem. From the figure,
Therefore, the most efficient triangular section is the 90° V-notch.