Cut and fill calculation: grid, cross-section and surface-to-surface

Cut and fill is the earthwork question: how much ground has to come out, how much has to go in, to get from the surface that exists to the surface the design wants. The arithmetic is a comparison of two surfaces. The three methods in use differ only in how completely they make that comparison, and the difference matters for the number you get, the time it takes, and how well you can defend it.

The one idea under all three methods

At any point on the site there is an existing ground height and a design height. Where existing is above design, the difference is cut. Where it is below, the difference is fill. Multiply each difference by the area it represents and add up: that is the earthwork. Every method is a way of choosing the points and the areas.

1. The grid method

Lay a square grid over the site, say 10 m by 10 m. At each grid corner, read the existing height from the survey and the design height from the plan. Each cell's volume is the average of its four corner differences times the cell area. Add the cells.

It is quick and it can be done by hand, which is why it is still taught. It is also coarse: a cell that is half cut and half fill averages to little or nothing, so the net is reasonable and the gross cut and fill, the figures that cost money, are understated. Finer grids fix that at the cost of more work, and a grid can never follow a ditch or a bund narrower than a cell.

2. The cross-section (average end area) method

Draw sections across the site at intervals, typically 20 to 50 ft or 10 to 25 m along a road or a channel. On each section, draw the existing ground and the design and measure the area of cut and the area of fill between them. The volume between two sections is the average of their end areas times the distance between them.

This is the method for linear work: roads, pipelines, levees, canals. It mirrors how such jobs are set out and paid, by chainage. Its weakness is between the sections, where it assumes the ground changes evenly, so a hump or a hollow between two sections is missed. Closer sections reduce the error; the prismoidal correction improves it for curved transitions. To run the numbers for your own sections, use the cut and fill calculator.

3. Surface-to-surface

Compare the two surfaces everywhere. With a drone survey, the existing ground is already a continuous grid of heights, often one every 2 to 5 cm, and the design is a surface (a triangulated LandXML or a DXF from the engineer). Software subtracts one from the other at every point and sums cut and fill separately, over the whole site or inside any boundary you draw.

There is no sampling, so nothing between sections or inside cells is missed, and the gross figures are right rather than averaged away. The whole site, however irregular, takes the same effort as one section. What it needs is a design surface in a form the software can read, and an existing-ground survey on the same datum, which is where the real work moves to.

Doing it from a drone survey

The sequence with a drone is short, and each step has one thing to get right.

1. Fly the existing ground with enough overlap and, for earthwork, ground control, RTK or PPK, so the survey sits on the same datum as the design. (What those give you.)
2. Process to a terrain model. Use the digital terrain model (DTM), not the surface model, where vegetation or equipment stood on the ground: the DTM has them removed. (DSM, DTM and which to use.)
3. Load the design surface as LandXML (the format every civil package exports) or a DXF with 3D faces, in the same coordinate system and vertical datum.
4. Draw the boundary of the area being priced and read cut, fill and net. In Clarity the cut and fill tool does this against a design or against an earlier flight, and prints the area measured with the figures.

Where cut and fill goes wrong

• A datum mismatch. The survey in ellipsoid heights, the design in orthometric; or two surveys from different control. A 1 m offset turns the whole site into fill. Clarity flags a whole-site height offset between two flights when it sees one, and lets you level them on ground you know did not change, but the cure is control shared between the survey and the design.
• A surface model instead of a terrain model. Trees and stockpiles in the "existing ground" become cut that nobody will ever excavate.
• The edge of the design. Outside the design's extent there is no design height. Software should stop at the edge and say how much of your boundary it measured, not assume zero.
• Bank, loose and compacted. Cut and fill are bank volumes. Trucks carry loose volume; the embankment is compacted. Apply the factors for the material, not to the survey.

A worked example

A building pad, 60 m by 40 m, on ground that falls 1.8 m across the long side. Design level is set at the mean. By the grid method with 10 m cells and heights read at 35 corners, cut comes to 1,120 m³ and fill to 1,080 m³. Surface-to-surface on a 3 cm drone terrain model over the same boundary gives cut 1,214 m³ and fill 1,166 m³: about 8% more of each, from the ground the grid averaged away. The net is nearly the same in both, 40 m³ against 48 m³. The contractor is paid on the gross.