Earthwork is measured by slicing the job into cross sections and putting the volume back together between them. Two formulas do almost all of it, and the choice between them is a choice about how much you measured.

Average end area: $V = \dfrac{L\,(A_1 + A_2)}{2}$, read aloud V equals L times A-one plus A-two, over two. $A_1$ and $A_2$ are the cross-section areas at the two stations, in square metres; subscript 1 is the near station and 2 the far one, though here the order does not matter because they are added. $L$ is the chainage between them in metres, and $V$ is the volume in cubic metres. It is a trapezoid extruded down the line: average the ends, drag the average the length of the run.

The prismoidal formula: $V = \dfrac{L\,(A_1 + 4A_m + A_2)}{6}$, where $A_m$ is the area of a section taken midway between the two ends, and $L$ is still the full distance end to end — not the half-distance to the middle section. This is Simpson's rule in a hard hat. The weights 1, 4, 1 sum to 6, which is why the six sits underneath, and the four on the middle section is the whole reason the formula is better.

What is the difference actually worth? Where the ground runs evenly between stations, almost nothing — and if the middle area happens to be the exact average of the ends, the two formulas give the identical answer. Where a section bulges or waists, they part company by a few percent, and end-area is the one that over-reports. That bias is why contractors have historically been content with it and why specifications for large works call for the prismoidal. On a hundred thousand cubic metres, a few percent is a week of trucks.

And keep cut and fill in separate runs. A section that is part cut and part fill has two areas, and letting them cancel inside one calculation loses both the excavation and the embankment at once.

Last habit, and this lesson opens with it every time: estimate before you compute. Round the two areas, average them in your head, multiply by the chainage. You will land inside the right order of magnitude in about two seconds, and every misplaced decimal point in the real calculation announces itself immediately.