The formula

For two cross-sections a distance L apart, with areas A₁ and A₂:

V = L × (A₁ + A₂) ÷ 2

Work out cut and fill separately. Never net them section by section: a section with 3 m² of cut and 2 m² of fill moves 5 m² of earth, not 1 m².

Worked example

Cross-sections every 20 m, with areas from the section calculation (see how to calculate the cut and fill area of a cross-section):

Chainage Cut area (m²) Fill area (m²)
0+000 6.20 0.00
0+020 7.45 0.30
0+040 5.10 1.85
0+060 2.40 4.60
0+080 0.95 6.75

Each 20 m length:

From To L (m) Cut volume (m³) Fill volume (m³)
0+000 0+020 20 (6.20 + 7.45) ÷ 2 × 20 = 136.50 (0.00 + 0.30) ÷ 2 × 20 = 3.00
0+020 0+040 20 (7.45 + 5.10) ÷ 2 × 20 = 125.50 (0.30 + 1.85) ÷ 2 × 20 = 21.50
0+040 0+060 20 (5.10 + 2.40) ÷ 2 × 20 = 75.00 (1.85 + 4.60) ÷ 2 × 20 = 64.50
0+060 0+080 20 (2.40 + 0.95) ÷ 2 × 20 = 33.50 (4.60 + 6.75) ÷ 2 × 20 = 113.50
Total 370.50 202.50

So this 80 m reach has 370.50 m³ of cut and 202.50 m³ of fill. If the spacing changes along the route, use the actual distance between each pair of sections for L.

When it over-estimates

Average end area assumes the earth between two sections is a prism whose shape changes evenly. Where a cutting or an embankment runs out to nothing, the solid is closer to a pyramid, and the formula gives too much:

  • Area 8.00 m² at one section, 0 at the next, 20 m apart.
  • Average end area: (8.00 + 0) ÷ 2 × 20 = 80.00 m³.
  • A pyramid of the same base and length: 8.00 × 20 ÷ 3 = 53.33 m³.

In practice the error is small on a long, even route and largest at the start and end of cuttings and fills. Two ways to reduce it:

  • Take sections closer together where the cut or fill is changing quickly, and at every point where the ground changes slope.
  • Use the prismoidal formula where the estimate calls for it: V = L ÷ 6 × (A₁ + 4 Aₘ + A₂), where Aₘ is the area of a section actually taken (or drawn) halfway between the two, not the average of A₁ and A₂.

Many estimates and contracts specify average end area for earthwork; check which method your department’s schedule asks for before choosing.

Checklist for an earthwork abstract

  • Cut and fill in separate columns, and separate rows for each layer (for example bed cut and each bank of a canal, or earth top and sub-base of a road).
  • The chainage and the distance L for every pair of sections, so anyone can check a line.
  • Totals per layer at the bottom, carried to the estimate.