Civil Engineering

Earthwork Volume Calculator

Compute the volume of cut or fill between two cross-sections using either the average end area method or the prismoidal formula. Enter the two end areas, the distance between them, and the swell and shrinkage percentages, and this calculator returns the bank, loose and compacted volumes, the mass of material, the swell factor and the number of truck loads required for the haul.

Average end area · prismoidal formula · swell and shrinkage factors

🛣 Cross-Sections and Material

Inputs convert when you switch. Results are always reported in SI — see the note under the results.

Volume Method

End Area A₁ (m²)

End Area A₂ (m²)

Mid-Section Area Aₘ (m²)

The area measured at the midpoint, not the average of the two ends. Only used by the prismoidal method.

Distance Between Sections L (m)

Swell (%)

Bulking on excavation. See the soil table below.

Shrinkage (%)

Reduction on compaction in the fill.

Bank Density (t/m³)

Loose Density (t/m³)

Truck Capacity (m³)

Heaped capacity. Trucks carry loose volume, not bank volume.

📊 Volumes and Haulage

Press Calculate Volumes to compute the earthwork.

ƒ Governing Formulae

Volume between two cross-sections

Average end area V = L (A₁ + A₂) / 2
Prismoidal V = L (A₁ + 4Aₘ + A₂) / 6
Prismoidal correction = V_prismoidal − V_average

State conversions

V_loose = V_bank (1 + swell)
V_compacted = V_bank (1 − shrinkage)
mass = V_bank × ρ_bank
swell factor = ρ_bank / ρ_loose = V_loose / V_bank
trucks = ⌈ V_loose / truck capacity ⌉

A₁, A₂ — cross-section areas at the two ends of the length

Aₘ — area measured at the midpoint, not the mean of the ends

L — distance along the chainage between the two sections

V_bank — volume in place, before excavation

V_loose — volume after excavation, once bulked

V_compacted — volume after placement and rolling

ρ_bank, ρ_loose — density in place and in the loose state

The swell percentage and the two densities are separate inputs, and nothing forces them to agree. The loose volume is driven by the swell percentage alone; the densities only produce the mass and the swell factor. Since ρ_bank/ρ_loose is mathematically the same thing as V_loose/V_bank, entering a density pair that implies a different swell to the one you typed is a contradiction. This page cross-checks the two and warns you if they disagree by more than two percent, but the calculation itself will not stop.

Aₘ is a measured area, not an average. If you set the mid-section area to the mean of the two ends, the prismoidal formula collapses to exactly the average end area result and the correction reads zero — which tells you nothing. The whole value of the prismoidal method is that the real mid-section usually is not the mean, and the correction quantifies how much the linear assumption was costing you.

What this does not cover. One pair of sections at a time — there is no chainage table, no mass-haul diagram, no free-haul or overhaul calculation, and no cut-and-fill balance across a scheme. Bulking of rock after blasting is highly variable and the single swell figure here is a crude stand-in. Truck loads are computed from volume alone, so where a dense material reaches the axle load limit before the body is full, the real count will be higher.

📝 Worked Example 1 — Average End Area

Given

A cutting between two chainages 50 m apart. The cross-section area is 120 m² at the first and 60 m² at the second. The material is common earth with 25 percent swell and 12 percent shrinkage, a bank density of 1.8 t/m³ and a loose density of 1.44 t/m³. Haulage is by 10 m³ trucks.

Step 1 — bank volume

V_bank = 50 × (120 + 60) / 2 = 50 × 90 = 4500.00 m³

Step 2 — loose volume for the haul

V_loose = 4500 × 1.25 = 5625.00 m³
This is the figure the trucks see. Using the bank volume here would undercount the haul by a quarter.

Step 3 — compacted volume in the fill

V_compacted = 4500 × 0.88 = 3960.00 m³
So 4500 m³ taken out of the cut builds only 3960 m³ of finished fill.

Step 4 — mass and haulage

mass = 4500 × 1.8 = 8100.00 t
swell factor = 1.8 / 1.44 = 1.250, which agrees with the 25 percent entered
trucks = ⌈5625 / 10⌉ = 563 loads

Bank volume 4500.00 m³ — what you excavate and get paid for

Loose volume 5625.00 m³ — what you haul

Compacted volume 3960.00 m³ — what you build

Mass 8100.00 t · Swell factor 1.250

Truck loads 563

These are the calculator's default inputs. Press Calculate Volumes without changing anything and you should get exactly these figures back. Three different cubic metre figures for the same material is the whole point of the exercise — 4500 in the ground, 5625 in the truck, 3960 in the embankment.

📏 Worked Example 2 — What the Prismoidal Correction Buys You

Given

The same cutting, but the surveyor has also measured the section at the midpoint and found it to be 80 m², not the 90 m² the average of the two ends would suggest. The ground is dished between the chainages.

Step 1 — the prismoidal volume

V = 50 × (120 + 4 × 80 + 60) / 6 = 50 × 500 / 6 = 4166.67 m³

Step 2 — the correction

Prismoidal correction = 4166.67 − 4500.00 = −333.33 m³
The average end area method overstated the cut by 7.4 percent.

Step 3 — what that means downstream

Loose volume falls from 5625.00 to 5208.33 m³
Truck loads fall from 563 to 521 — forty two fewer journeys
Mass falls from 8100.00 to 7500.00 t

Average end area 4500.00 m³, 563 loads

Prismoidal 4166.67 m³, 521 loads

Correction −333.33 m³, a 7.4 percent overestimate avoided

Which way does the error go?

Average end area overestimates whenever the mid-section is smaller than the mean of the ends, which is the usual case for a cutting through a rounded hill or a fill across a valley. It underestimates when the mid-section is larger. The error grows with the spacing of the sections and with how sharply the ground curves, so on rolling terrain surveyed at 50 m intervals the correction is well worth having. On flat ground with sections every 20 m it is negligible.

The mid-section area must be measured. Enter 80 here and you get a real correction; enter 90, the mean of the two ends, and the prismoidal formula returns exactly the average end area answer with a correction of zero. If Aₘ is left at zero the volume comes out at one third of the correct value, so this page blocks that rather than letting it through silently.

📐 Inputs, Units and Accepted Ranges

Input	SI unit	Imperial unit	Accepted	Notes
Volume method	—	—	average end area, prismoidal	Prismoidal needs the mid-section area
End areas A₁, A₂	m²	ft²	≥ 0	One may be zero where the section runs out
Mid-section area Aₘ	m²	ft²	> 0 for prismoidal	Measured at the midpoint, not averaged
Distance L	m	ft	> 0	Along the chainage between sections
Swell	%	%	≥ 0, typically 10 – 80	Should agree with the density pair
Shrinkage	%	%	≥ 0, typically 5 – 20	Values above 100 would give a negative volume
Bank density	t/m³	pcf	> 0, typically 1.4 – 2.4	Density in place
Loose density	t/m³	pcf	> 0, less than bank	Density after excavation
Truck capacity	m³	yd³	> 0	Heaped capacity, loose measure

📚 Reference Tables

Swell, shrinkage and density by material

Material	Swell, %	Shrinkage, %	Bank density, t/m³	Loose density, t/m³
Sand, dry	10 – 12	5 – 8	1.60	1.45
Sand and gravel	12 – 15	8 – 10	1.90	1.68
Common earth / loam	20 – 25	10 – 12	1.80	1.44
Clay, stiff	25 – 30	12 – 15	2.00	1.58
Clay, hard / shale	30 – 40	15 – 20	2.20	1.65
Rock, well blasted	50 – 60	—	2.60	1.70
Rock, poorly blasted	60 – 80	—	2.60	1.50

The density pairs in this table are consistent with the swell figures beside them — loose density is bank density divided by one plus the swell. Rock is not normally quoted a shrinkage because it is placed as rockfill rather than compacted to a soil density.

Swell factor and load factor — two names for reciprocal quantities

Swell, %	Swell factor V_loose/V_bank	Load factor V_bank/V_loose	1 m³ bank becomes
10	1.10	0.909	1.10 m³ loose
15	1.15	0.870	1.15 m³ loose
20	1.20	0.833	1.20 m³ loose
25	1.25	0.800	1.25 m³ loose
30	1.30	0.769	1.30 m³ loose
50	1.50	0.667	1.50 m³ loose
80	1.80	0.556	1.80 m³ loose

The two are easily confused because both are called "factor". This calculator reports the swell factor, greater than one. If your reference quotes a load factor of 0.80, that is the same material as a swell factor of 1.25.

How much the prismoidal correction matters — A₁ = 120, A₂ = 60, L = 50 m

Measured Aₘ, m²	Prismoidal V, m³	Correction, m³	Error in average end area
70	3833.33	−666.67	+17.4 % overestimate
80	4166.67	−333.33	+8.0 % overestimate
90 (the mean)	4500.00	0.00	exact
100	4833.33	+333.33	−6.9 % underestimate
110	5166.67	+666.67	−12.9 % underestimate

Typical truck and plant capacities

Vehicle	Heaped capacity, m³	Heaped capacity, yd³	Loads for 5625 m³ loose
Small tipper, 6 wheeler	6	7.8	938
Standard tipper, 10 wheeler	10	13.1	563
Large tipper, 12 wheeler	16	20.9	352
Articulated dump truck, 30 t	18	23.5	313
Rigid dump truck, 40 t	24	31.4	235

Check the payload mass as well as the volume. At a loose density of 1.44 t/m³ a 10 m³ body carries 14.4 t, which is within a 10 wheeler's limit; the same body full of blasted rock at 1.70 t/m³ carries 17 t and may not be.

❓ Frequently Asked Questions

What is the difference between the average end area and prismoidal methods?

The average end area method takes the mean of the two end cross-section areas and multiplies by the distance between them, which assumes the section varies linearly along the length. The prismoidal formula weights the mid-section area four times against the two ends and divides by six, which is exact for any shape whose area varies as a quadratic. Average end area overestimates the volume whenever the ground is concave in the direction of the chainage, and the prismoidal correction is the difference between the two.

What is the difference between bank, loose and compacted volume?

Bank volume is the material in place before excavation, and it is what you measure from cross-sections and what you are usually paid for. Loose volume is the same material after excavation, swollen by the void space introduced when it is broken up, and it is what determines how many truck loads you need. Compacted volume is the material after placement and rolling in a fill, which is smaller than bank volume for most soils. One cubic metre in the cut is not one cubic metre in the truck and not one cubic metre in the fill.

What is the swell factor for common soils?

Sand and gravel typically swell by ten to fifteen percent, common earth and loam by about twenty to twenty five percent, clay by twenty five to forty percent, and blasted rock by fifty to eighty percent. Swell depends on how tightly the material was packed in place, so dense well graded soils swell more than loose ones. The corresponding load factor, which is bank volume divided by loose volume, runs from about zero point eight seven for sand down to around zero point six for rock.

How do you calculate the number of truck loads for an excavation?

Divide the loose volume by the heaped capacity of the truck and round up, because trucks carry loose material rather than bank material. A frequent error is dividing the bank volume instead, which underestimates the haul by the swell percentage. Also check whether the payload mass rather than the volume governs, since a dense material can reach the axle limit of the truck before the body is full.

Why does compacted volume come out less than bank volume?

Compaction by rolling forces the soil particles into a denser packing than they occupied in the natural ground, so a given mass of material occupies less space in the finished fill than it did in the cut. The reduction, called shrinkage, is typically five to ten percent for sand and gravel and ten to twenty percent for clay. It means the cut volume needed to build a given fill is larger than the fill volume, which is why a scheme that balances on paper often runs short of material.

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Earthwork Volume Calculator — multicalci.com. Computes one pair of cross-sections at a time. No mass-haul diagram, no overhaul, no cut-and-fill balance. Swell, shrinkage and density figures vary widely with material and compaction, so use site-specific test values for a tender. Truck loads are computed from volume only and may be limited by payload mass instead.