Trapezoidal Channel Flow Area

The flow area of a trapezoidal channel is the rectangle under the water, bottom width $B$ times depth $d$, plus the two sloping side wedges, which together come to $z\,d^{2}$. The side slope $z$ is the horizontal run for each 1 of rise, so a slope of 1 vertical to 3 horizontal is $z = 3$. Two special cases fall out: a triangular channel is $B = 0$ and a rectangular one is $z = 0$. The depth is the maximum flow depth, measured from the bed. This is plain geometry, so the relation is exact for the section you describe; the only approximation is how well a real ditch matches a trapezoid.

The relation is one of three section properties that the Federal Highway Administration's urban drainage manual (HEC-22, fourth edition, equations 6.7 to 6.9) lists for roadside and median ditches. The other two are the wetted perimeter, $P = B + 2d\sqrt{z^{2}+1}$, and the top width, $T = B + 2zd$. The page shows both, with the hydraulic radius $R = A/P$, in the note with each answer, because those are what Manning's equation, the Froude number, and the shear stress and bend checks take next. The equations take the same form in feet as in metres. When the area is known instead, the page solves for the bottom width, the side slope, or the depth, which is the positive root of $z\,d^{2} + B\,d - A = 0$.

Read the side slope as a design choice, not a measurement. The manual sets side slopes no steeper than the angle of repose of the soil or lining, and beside a road often 1V:3H or flatter so that an errant vehicle can cross them. Its own example is a channel with $B = 2.6$ ft, $z = 3$ and $d = 1.6$ ft, which has $A = 11.8$ ft², $P = 12.7$ ft and $R = 0.93$ ft, and with $n = 0.030$ and a slope of 0.01 carries 55.7 ft³/s. The exact area is 11.84 ft², and the manual prints 11.8, rounded to 0.1 ft².

Working backwards shows how rounding spreads. Putting the printed 11.8 ft² with $B = 2.6$ ft and $z = 3$ into the depth formula gives 1.5967 ft against the printed 1.6 ft, a difference that comes from the rounded area. The page's own example takes the same channel in metres, where the manual prints the dimensions but not the area: $B = 0.8$ m, $d = 0.5$ m and $z = 3$ give $A = 0.8 \times 0.5 + 3 \times 0.25 = 1.15$ m².