When solving problems in Engineering School (Cal Poly) involving a lagoon in this shape I was told to approximate the volume with a trapezoidal prism. Although the shape is trapezoidal it is not a prism, so this always bothered me a bit. After using two methods (1) breaking the shape down into simpler polyhedra with known volume formulas such as prisms and pyramids and (2) using calculus to integrate the cross-sectional area as a function of height I have the “gold bar formula.”
V = (1/6)(2LW+2lw+Lw+lW)h
Where L and W are the length and width of the large base, l and w are the length and width of the small base and h is the height from base to base. Notice that in the case of a rectangular prism (L=l and W=w) the formula reduces to V=lwh, and in the case of a pyramid (l = w = 0) the formula reduces to V=(1/3)LWh.
This formula has applications not only for man made water bodies, and gold bars, but chocolate, pottery, and decorative containers.