The method in one line
Take the cross section area at one station and the area at the next, average them, multiply by the distance between. That is the volume of the prism between the two sections. Do it for every pair along the run and add them up. It is the trapezoid rule applied to dirt, and it is the method most earthwork quantities on a linear job are still computed with because it is transparent and anybody can check it by hand.
Cut and fill are carried separately all the way through and only netted at the end. A section can be cut on one side of a road and fill on the other, and a job that nets to zero can still have three weeks of machine time in it. Netting early hides that.
Where it goes wrong, and by how much
The method assumes the section changes linearly between stations. Where a section runs out to nothing — the end of a cut, the start of a fill, a ditch that stops — average end area treats the shape as a wedge and gives half the end area times the length. The real shape is closer to a pyramid, which is a third. Average end area is 50 percent high on every one of those segments.
That matters more than it sounds. A 3.6 square foot section tapering to zero over 50 feet is 90 cubic feet by average end area and 60 by the pyramid formula: 3.33 versus 2.22 cubic yards on one segment. On a corridor with a dozen taper-outs it is enough to change what gets ordered. The other systematic error runs the same direction: average end area is high whenever the section is convex between stations, which is most of the time, and that is what the prismoidal correction exists to remove. The practical answer to both is closer sections where the shape changes, not more decimal places on the ones you have.
Bank, loose and compacted are three different yards
| Measure | What it is | Where it shows up |
|---|---|---|
| Bank | In place, undisturbed, before anything touched it | Cut quantities, what the sections actually measure |
| Loose | After it is broken up and swelled | Truck loads, stockpile size, what a hauler bills |
| Compacted | Placed and rolled to the required density | Fill quantities, what the specification accepts |
One bank yard of cut does not make one compacted yard of fill. With 12 percent shrink it makes 0.88, so 39.6 yards of fill needs about 45 yards of cut to produce it. Skip that step and a job that looks balanced on the section sheet arrives short. Swell works the other way and only affects hauling: the same bank yard becomes 1.25 loose yards on the truck, which is what fills the box and what gets paid for.
The balance point is where the money is
Running the difference between cumulative cut and cumulative fill along the line gives a curve, and where it crosses zero, everything dug so far exactly makes everything placed so far. That is the point material stops wanting to travel forward and starts wanting to travel back. It sets haul direction, it sets where a scraper turns, and on a long job the haul distance is a bigger number than the yardage. A quantity sheet that gives only totals says nothing about it.
The end areas themselves have to come from somewhere. The side-slope part of a section falls out of the slope stake calculator, the finished-surface depths from the grade stake calculator, and the elevations behind all of it from a checked levelling loop.
Questions people ask
Why is average end area wrong at a taper?
Because a section that goes from an area A down to zero over a length L is a pyramid, not a wedge. A pyramid is one third of the base area times the length; average end area gives you half, since it averages A and zero to get A over 2. That is a 50 percent overstatement on every segment that runs out to nothing, and it is systematic rather than random — it never errs low. The default here uses the pyramid form on those segments, with the conventional treatment available if a specification requires it.
What is the prismoidal correction and why is it not here?
It is a refinement that uses a third area at the middle of the segment to fit a curved solid rather than a straight-sided one, and it corrects the bias average end area has when the section changes shape rather than just size. It needs a middle section, measured or computed, that most quantity sheets do not carry. Rather than fabricate one, this page handles the largest and most predictable part of the same bias — the taper case — and leaves the rest to closer station spacing, which fixes it directly.
Do the stations have to be evenly spaced?
No. Each segment uses its own length, so uneven spacing is fine and often better: put sections close together where the shape changes fast and further apart on a stretch where nothing happens. What the stations do have to be is in increasing order, since the length of each segment comes from the difference between consecutive stations. Out-of-order lines are rejected rather than silently producing a negative volume.
What swell and shrink should I use?
Numbers agreed with the people doing the work, which is why they are inputs. Swell depends on the material and how it is dug; shrink depends on the material and the compaction required. Sand, clay and weathered rock behave differently, the same soil behaves differently wet, and a specification that demands high density shrinks more than one that does not. A hauler and an earthwork contractor will each have a figure they work to, and those figures are worth more than any table.
Does a balanced job mean no hauling cost?
No, and that is the most expensive misreading of a quantity sheet. Balanced means the total cut equals the total fill needed, so nothing has to leave or arrive. Every yard still has to be dug, moved, placed and compacted, and how far it moves is set by where the surpluses and shortfalls sit along the line rather than by the totals. A job that balances with all the cut at one end and all the fill at the other is far more expensive than one that balances in short hops, which is what the running balance and the balance point are for.