Earthwork volume using a middle cross-section
A middle cross-section tells you more about the shape than two ends alone. The prismoidal rule gives the middle area a weight of four before forming the volume.
What the formula is saying
Use the two end areas A₁ and A₂ plus the area Am halfway between them. Form A₁ + 4Am + A₂, divide by six, and multiply by the full end-to-end length.
Read the symbols in plain language
- L
- Section spacingm
- A₁
- Area 1m²
- Am
- Mid-aream²
- A₂
- Area 2m²
Sort out the units first
All areas use m² and L uses m. Am must be measured at the halfway station, not at any convenient intermediate point.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
- L · Section spacing
- 12 m
- A₁ · Area 1
- 10 m²
- Am · Mid-area
- 16 m²
- A₂ · Area 2
- 22 m²
Form the weighted area sum
The weights are 1, 4 and 1; they add to six.
(10) + 4 × (16) + (22) = 96 m²Divide by the total weight
This gives the Simpson-weighted average area.
(96) ÷ 6 = 16 m²Multiply by the full spacing
Use the distance between the two end sections.
(16) × (12) = 192 m³
Does this worked answer make sense?
With all three areas equal to A, the rule becomes AL. In the example the weighted mean is 16 m², giving 192 m³.
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Use these new values. Work it out first, then check your answer.
- L · Section spacing
- 18 m
- A₁ · Area 1
- 8 m²
- Am · Mid-area
- 12 m²
- A₂ · Area 2
- 20 m²
Find: Earthwork volume using a middle cross-section
A hint, not the answer
Use the two end areas A₁ and A₂ plus the area Am halfway between them. Form A₁ + 4Am + A₂, divide by six, and multiply by the full end-to-end length.
All areas use m² and L uses m. Am must be measured at the halfway station, not at any convenient intermediate point.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Form the weighted area sum
The weights are 1, 4 and 1; they add to six.
(8) + 4 × (12) + (20) = 76 m²Divide by the total weight
This gives the Simpson-weighted average area.
(76) ÷ 6 ≈ 12.66666667 m²Multiply by the full spacing
Use the distance between the two end sections.
(12.66666667) × (18) = 228 m³
Avoid the common trap
Do not use half the spacing for L. Do not invent the middle area by averaging the ends when the actual middle shape is unknown.
When this method applies — and when it does not
Prismoidal geometry or Simpson integration of smoothly varying cross-sectional area at equally spaced stations. Not universally exact for irregular terrain.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Lesson updated: · Worked examples checked against the implemented formula; not an independent engineering certification.