8.5 Earthwork Volumes, Contours, and Mass Diagrams

Average End Area and Prismoidal Methods

Earthwork volume is in the FS Survey Computations area because surveyors support grading, roadway, and site quantity work. Volume between two cross-sections taken L feet apart is computed from the cut (or fill) area at each section.

The average end area (AEA) method is the workhorse:

V = (L/2)(A1 + A2) (cubic feet) → divide by 27 for cubic yards.

The prismoidal formula is more accurate when the section shape changes between the ends, using a middle area Am:

V = (L/6)(A1 + 4·Am + A2)

The critical subtlety: Am is computed from the average of corresponding linear dimensions (averaged widths and heights) of the two end sections, not from the average of the two areas. Averaging the areas just reproduces AEA. Because AEA assumes a linear change in area while the true solid tapers as a prismoid, AEA generally overestimates the volume — often by a few percent on transitioning sections, more where one section is much larger than the other.

Worked Volumes and Unit Conversion

AEA example: Cross-section cut areas A1 = 80 ft² and A2 = 140 ft² at sections L = 100 ft apart. V = (100/2)(80 + 140) = 50 × 220 = 11,000 ft³ → 11,000 / 27 = 407.4 yd³.

Prismoidal example: With a middle area Am = 108 ft² (from averaged dimensions): V = (100/6)(80 + 4·108 + 140) = (16.667)(80 + 432 + 140) = 16.667 × 652 = 10,867 ft³ → 402.5 yd³. The AEA value (407.4 yd³) is about 1.2% larger, illustrating its tendency to overestimate.

Unit discipline

Volumes are quoted in cubic yards in U.S. earthwork. Remember 1 yd³ = 27 ft³ (3 ft × 3 ft × 3 ft). Convert at the very end, not mid-calculation. A frequent FS trap is dividing by 3 instead of 27, or by 9 (which is ft²→yd²).

Contours, Grids, and the Mass-Haul Diagram

Where cross-sections are unavailable, volume can be approximated from a borrow-pit grid (sum of corner elevation differences from a datum, weighted by how many cells each corner touches) or from contour areas combined by the AEA or prismoidal formula using the contour interval as L. These give site stockpile or excavation quantities directly from a topographic surface.

For linear projects, the mass-haul (mass) diagram is the key planning tool. It plots cumulative earthwork volume (cut counted positive, fill negative, after applying a shrinkage/swell factor) on the vertical axis against station on the horizontal axis. Reading it:

• A rising segment is net cut; a falling segment is net fill.
• A horizontal line crossing the curve (a balance line) marks stations between which cut exactly balances fill.
• Where the curve returns to a previous level, the material between those stations is balanced and the haul direction is set by whether the loop is above or below the balance line.
• A curve ending above its start means surplus material (waste); ending below means a deficit requiring borrow.

Cut/fill sign control

Net volume = total cut − total fill, in consistent units after the shrinkage factor (compacted fill needs more bank volume than its in-place measure). Mixing cut and fill signs, or forgetting the shrinkage factor, corrupts the borrow/waste decision — the practical reason the exam stresses sign discipline over raw formula recall.

Shrinkage, Swell, and the Borrow-Pit Grid

Earthwork volumes are quoted in three states, and the FS exam expects candidates to keep them straight:

Swell is the volume increase when soil is excavated and loosened (loose > bank); it governs hauling because trucks carry loose volume. Shrinkage is the volume decrease when soil is compacted into fill (compacted < bank); it governs how much bank material is needed to build a given fill. A typical shrinkage factor of, say, 0.90 means 1.0 yd³ of compacted fill requires about 1.0/0.90 ≈ 1.11 yd³ of bank excavation. Applying the shrinkage factor before balancing cut and fill is essential; ignoring it understates the borrow needed.

Borrow-pit (grid) volume

For stockpiles and excavations, the borrow-pit method overlays a square grid and reads the cut/fill depth at each grid corner. The volume is:

V = (grid-cell area / 4) × Σ(hₙ × n)

where each corner height h is weighted by n, the number of cells touching it: corner = 1, edge = 2, interior = 4. For example, on a 50-ft × 50-ft grid, a corner used by one cell counts its depth once; an interior node shared by four cells counts four times. Summing the weighted depths and multiplying by (cell area ÷ 4) gives the excavation volume directly from the surveyed surface — a method that connects topographic field data straight to a construction quantity without cross-sections.