Triangular-channel hydrostatic pressure head ($I=AH/3,I_x/A=H_x$)

For a fixed symmetric triangular channel with depth $H$ and area $A=k{H}^2$, the hydrostatic first moment is $I={\int }_A(H-z)dA=k{H}^3/3=AH/3$. Since $dI/dH=A$, the depth-averaged pressure gradient is $g{I}_x/A=g{H}_x$. A mean geometric depth $H/2$ cannot replace $H$ in that gradient without an extra factor two. This distinction prevents confusion between pressure head, centroid depth, and hydraulic radius.