Earthwork Volume by the Prismoidal Formula

The prismoidal formula is exact for any solid whose cross-sectional area varies as a quadratic along its length, which covers the wedges, prisms, pyramids and transition sections that make up almost all real earthwork. It is Simpson's rule with a surveyor's vocabulary, and the 1-4-1 weighting is what makes it exact where the trapezoidal average end area is merely close. Note the subtlety that catches everyone: A_m is the area of the section physically measured at the midpoint, not the average of A₁ and A₂. Substitute the average and the whole thing algebraically collapses back into the end area method.

A worked example: end areas of 120 ft² and 180 ft² one station apart, with the section actually taken at 0+50 measuring 140 ft². V = 100 × (120 + 4 × 140 + 180)/6 = 14,333 ft³ = 530.9 yd³, some 4.4 % less than the 555.6 yd³ the end area method returns. That difference, the prismoidal correction, is why large-quantity contracts and borrow pit pay quantities specify this method: on a million-yard job, four percent is the profit.