The hook
Earthwork is the most expensive line item on most civil projects, and surveyors compute it. Two formulas: average end area (fast, slightly wrong) and the prismoidal formula(slower, exact for prismoids). The exam will give you cross-section areas at stations and ask for volume — pick the right tool.
Memorize these
Concepts that show up on the exam
End area
Cross-sectional area of the cut or fill at a single station. Computed by the coordinates method (FS 5.E) on the stationing-perpendicular section.
Average end area (AEA)
Volume between two stations = average of the two end areas times the length between them. Easy, slightly conservative for warped sections.
Prismoidal formula
Exact for any prismoid. Uses end areas + a third area at the midpoint. V = L/6 × (A₁ + 4Aₘ + A₂).
Mass diagram
Plot of cumulative cut/fill earthwork along a route. Used to plan haul distances and balance points (where total cut = total fill so far).
Bulking / shrinkage
Excavated soil expands ("bulks") then compacts back smaller than original ("shrinks"). Earthwork tables apply factors of typically 1.10–1.30 (bulk) and 0.85–0.95 (shrink). NOT in either volume formula.
Borrow / waste
Borrow = bringing in fill from off-site. Waste = hauling out excess cut. The mass diagram tells you how far each unit travels; haul cost = volume × distance.
Keep these in muscle memory
Formulas to know cold
Average end area
V = L · (A₁ + A₂) / 2L = length between sectionsA = end areas
Prismoidal
V = (L / 6) · (A₁ + 4Aₘ + A₂)Aₘ = cross-section at midpoint of L
Pyramid (transition section)
V = (L · A) / 3When one end area = 0 (transition from cut to fill), use the pyramid formula, not AEA.
Try it before you peek
Worked example
The problem
A roadway cut between Sta 10+00 and Sta 11+00 has end areas A₁ = 320 sq ft and A₂ = 180 sq ft. Compute the volume by (a) average end area, (b) prismoidal if Aₘ = 244 sq ft at Sta 10+50, and (c) the percent difference.
Don't fall for these
What trips people up
Forgetting to convert ft³ to yd³
Earthwork is invoiced in cubic YARDS. 1 cu yd = 27 cu ft. Surveyors who deliver volumes in cubic feet to a contractor will get a phone call.
AEA at transition (cut↔fill) sections
When one end area is zero (the section transitions from cut to fill), AEA gives
V = L·A₁/2, but the actual shape is a pyramid: V = L·A₁/3. AEA is 50% too high here.Mixing cut and fill in the same volume
Cut and fill are tracked SEPARATELY. A station can have both. Sum cut volumes and fill volumes independently; don\'t let them cancel in the math.
Test yourself
How well did it stick?
A quick 5-question check on Volume. See where you stand and what to review.