Areas and Volumes

A cross-section is levelled at intervals, each ordinate being the height difference between existing ground and the design formation. The area between them, integrated along the route, becomes cubic metres — and cubic metres are what the contractor is paid for.

The trapezoidal error is systematic — it always misses the same way, so taking more sections narrows nothing. Simpson's 1-4-2-4-1 weighting is not arbitrary; it is what makes the parabola fit exactly.

The same two rules apply one dimension up: the end-area method is trapezoidal and the prismoidal rule is Simpson's. End-area over-reads when the section changes shape between chainages, and the prismoidal correction is the difference — small per section, and worth real money over a long earthwork.

Where the shape is irregular rather than linear — a reservoir, a spoil heap — volumes come from contour areas: the volume between two contours is their mean area times the interval, summed layer by layer.

Plotting cumulative cut minus fill along a route gives the mass haul curve. Where it returns to a horizontal balancing line, cut and fill between those chainages cancel exactly — so material is moved once, over the shortest haul.

Earthmoving is priced per cubic metre per kilometre, which is why the diagram is drawn before a machine moves. Rising means cut, falling means fill, and the area between the curve and the balancing line is the haul.

Simpson's rule is exact for the profile it is given — and if sections are 50 m apart on undulating ground, it is exact arithmetic on a poor sample. The dominant error is the sampling interval, decided in the field. Quoting a volume to six significant figures from sparse sections is arithmetic theatre.

Laser scanning and drone photogrammetry capture millions of points rather than hundreds, so volumes come from a triangulated surface and the sampling error largely disappears. What has not changed is the need for surveyed ground control, without which the whole cloud sits in the wrong place.