How the problems are made
Each set invents a stretch of road: a template with its half width and side slopes, ground heights at each station, and from them real three-level, five-level, irregular and side-hill cross sections. Every area comes from the corner coordinates, every volume from those areas, and the mass ordinates from those volumes, so the numbers hang together from the first section to the last haul. Heights are booked to 0.1 ft or 0.01 m, and a problem is only used when no rounding lands exactly on a half.
Example: balance points and overhaul
In set MASS25 (level 5, US units, 600 ft free haul) the ordinate falls from +1,448 yd³ at 20+00 to −43 at 21+00, so the balance point is 20+00 + 100 × 1,448 ÷ 1,491 = 20+97.1. The second balance runs on to 30+00 and is longer than the free haul: a 600 ft chord fits it at −1,368 yd³, from 22+41.2 to 28+41.2. That leaves 2,275 yd³ of free haul and 1,368 yd³ of overhaul. The area outside the chord ends is 1,955 sta-yd, so the centres of gravity are 743 ft apart, 143 ft beyond the free haul. Open set MASS25
What each level asks
| Level | You are given | You find |
|---|---|---|
| 1 Sections | Two drawn cross sections with slope stake notes | Slope stake offsets, cut and fill areas by coordinates |
| 2 End areas | A profile and the end areas at each station | Cut and fill volumes between stations, totals |
| 3 Prismoidal | Three-level sections in cut and in fill | End-area volumes, prismoidal corrections, shrinkage, spare cut or shortage |
| 4 Mass ordinates | Cut and fill volumes and the shrinkage | Adjusted volumes, net volumes, mass ordinates |
| 5 Mass diagram | The mass ordinates, drawn | Balance points, haul, average haul, free haul chord, overhaul |
| 6 Borrow and waste | End areas and the shrinkage | The whole table, balance points, borrow or waste at the end |
Why the agency rules are settings
Agencies and textbooks disagree on three things, so each is a choice printed on every sheet and in the key. NDDOT adds shrinkage to the fill and takes the project average haul as the free haul; Montana DOT does not use free haul at all, and WYDOT works out haul only for estimating. Some course notes take shrinkage off the cut. Where one end area is zero, WYDOT and NDDOT keep the average end area while the LibreTexts transportation text uses a pyramid. These are practice problems, not a pay quantity calculation for a contract.
Questions
How do you calculate earthwork volume by the average end area method?
Average the end areas of two cross sections and multiply by the distance between them: V = (A₁ + A₂) ÷ 2 × L. In US units divide cubic feet by 27 for cubic yards. Fill areas of 100 and 120 ft² 100 ft apart give 220 ÷ 2 × 100 ÷ 27 = 407 yd³.
When do you use the prismoidal formula instead?
The prismoidal formula, V = L (A₁ + 4Am + A₂) ÷ 6, is closer when the section changes shape between stations. For three-level sections it comes down to a correction on the end-area volume: Cp = L (c₁ − c₂)(D₁ − D₂) ÷ 12, where c is the centre height and D the width between slope stakes. Subtract Cp, keeping its sign.
What is a mass haul diagram?
A running total of cut minus fill, after shrinkage, plotted against station. It rises through cut and falls through fill. Where it crosses the balance line, cut and fill balance; the area between the curve and the line is the haul, in station-yards or cubic-metre stations.
How do you find a balance point?
Find two stations whose mass ordinates have opposite signs and interpolate: station + L × M₁ ÷ (M₁ − M₂). With +1,448 at 20+00 and −43 at 21+00, the balance point is 20+00 + 100 × 1,448 ÷ 1,491 = 20+97.1.
What are free haul and overhaul?
Free haul is the distance earth is moved with no extra payment. On the mass diagram it is a horizontal chord of that length touching the curve. Volume between the chord and the peak (or valley) is free haul; volume between the chord and the balance line is overhauled, and overhaul = that volume × (distance between centres of gravity − free haul).
Is shrinkage applied to the cut or to the fill?
Both are taught. NDDOT and WYDOT add it to the fill: with 20% shrinkage, fill × 1.2 balances against bank cut. Some course notes take it off the cut instead: cut × (1 − s). The mass ordinates are then in bank or compacted measure. Pick the rule your course uses; it is printed on every sheet.
What does the end of the mass diagram tell you?
If the last ordinate is above zero, cut is left over and goes to waste. If it is below zero, the fill needs more than the cut gives, and the difference comes from borrow.
Sources, checked October 26, 2026: LibreTexts, Fundamentals of Transportation 7.3 Earthwork (CC BY-SA): coordinate areas, average end area, pyramid and prismoidal formulas, worked volumes; NDDOT, Earthwork and Mass Diagrams (2011, PDF): shrinkage on embankment, average haul, overhaul; WYDOT Road Design Manual 3-03, Earthwork Design (2011, PDF): end areas, compaction factors, mass ordinates, balance points, haul; Montana DOT, Earthworks Procedures (PDF): cut and fill factors, balance points, free haul not used; Mustansiriyah University, Mass Haul Diagram lecture (PDF): shrinkage taken off the cut; Autodesk Civil 3D help: free haul chord and overhaul on a mass haul diagram; Iowa DOT Standard Specifications 2108, Overhaul (2012, PDF): free haul limits fixed on the mass diagram, overhaul in station-yards or station-metres.