Cut and Fill

Earthwork Volumes from Two Surfaces

CE 414 Engineering Applications of GIS
Civil & Construction Engineering
Brigham Young University
Dr. Dan Ames

CE 414 · Week 10 — Cut and FillLast Updated: 2026-10-09© 2026 Daniel P. Ames · CC BY 4.0

Today's Goals

By the end of class you should be able to:

  • Compute cut and fill from an existing and a design surface, cell by cell
  • Find the balance grade of a level pad
  • Run Cut Fill in ArcGIS Pro and read its VOLUME sign
  • Say what a grid volume leaves out: side slopes, swell and shrink, DEM error
CE 414 · Week 10 — Cut and Fill

Part 1 — Earthwork Is a Volume

CE 414 · Week 10 — Cut and Fill

Two Surfaces

  • Existing ground: the DEM
  • Design surface: the pad, the road, the channel, as built
  • Cut: ground above the design, dug out
  • Fill: ground below the design, built up
CE 414 · Week 10 — Cut and Fill

The Grid Method

  • Every cell: Δz = ground − design
  • Cell volume = Δz × cell area (100 m² for a 10 m cell)
  • Cut = sum of the positive cells; fill = sum of the negative ones
  • Units: project first, so the cell size and z are both meters
CE 414 · Week 10 — Cut and Fill

Your Turn — Balance a Tiny Pad

Nine 10 m cells of existing ground (m):
102 103 105
101 102 104
100 101 103
  1. Level the pad at 102 m. Cut? Fill? (m³)
  2. At what grade do cut and fill balance?
CE 414 · Week 10 — Cut and Fill

The Balance Grade

  • Raise the pad: less cut, more fill
  • Balance: cut = fill, nothing trucked on or off
  • For a level pad on a grid, balance is the mean ground height under it
  • Our pad: 1,517.7 m, 21.9 thousand m³ each way
CE 414 · Week 10 — Cut and Fill

Cut Fill in ArcGIS Pro

  • Inputs: the before and after surfaces, same cells
  • Output raster: one region per change, with VOLUME and AREA
  • VOLUME > 0: material removed (cut) · < 0: material added (fill)
  • Our pad at balance: +21,852.5 and −21,852.5 m³, 7,500 m² each
CE 414 · Week 10 — Cut and Fill

Roads: Average End Area

center

  • Highway earthwork is computed from cross-sections at stations
  • Between two sections: V ≈ L × (A₁ + A₂) ÷ 2
CE 414 · Week 10 — Cut and Fill

Part 2 — What the Volume Leaves Out

CE 414 · Week 10 — Cut and Fill

A Grid Volume Is a First Estimate

  • Side slopes: a pad needs cut and fill slopes at its edges
  • Swell and shrink: dug soil takes more room loose, less when compacted
  • Unusable material: topsoil and rock do not go back as fill
  • The DEM itself: a 10 m cell averages the ground; buildings and roads are in it
CE 414 · Week 10 — Cut and Fill

Thursday: A Volume You Cannot Dig

  • Same arithmetic, bigger question: how much rock is in a mountain?
  • The "design surface" is the ground under the mountain, which nobody can see
  • That is Lab 9 — Big Southern Butte
CE 414 · Week 10 — Cut and Fill

Final Thoughts

Quiz: Cut or Fill? Five questions on cut, fill, balance, and the Cut Fill tool. Scan the code, or open the link under it.

  • Not graded, nothing recorded: a quick check that today's material made sense
  • Note the sign question; it is a common mistake in the report

Before next class

center

byu-hydroinformatics.github.io/ce414-gis-applications/quizzes/earthwork/

CE 414 · Week 10 — Cut and Fill

Tuesday of Week 10. Thursday is volumes with rasters and Lab 9 (Big Southern Butte); today is the same idea in the form every civil engineer meets first: how much soil a design moves. The title image is a hypothetical 150 m x 100 m level pad placed on the real Lab 5 DEM of Provo's east bench, colored by how far the ground must come down (red, cut) or go up (blue, fill).

stamp:begin

stamp:end

A long section through the pad, along its middle row of cells. The design surface equals the ground outside the pad. The pad's ends are vertical here because the design has no side slopes; a real grading plan ties the pad back to the ground with slopes (for example 2:1 or 3:1), which adds earthwork the simple grid misses.

The same arithmetic as Week 7's lake volumes and Lab 9's butte: a difference of two surfaces, multiplied by the cell area, summed. The map is the pad at its balance grade: red cells are cut, blue are fill; the bench slopes about 8 % down to the west.

A made-up grid for hand practice. Answers: at 102 m the differences are 0, 1, 3 / -1, 0, 2 / -2, -1, 1, so cut = (1 + 3 + 2 + 1) x 100 = 700 m³, fill = (1 + 2 + 1) x 100 = 400 m³, and 300 m³ must leave the site. Balance is the mean of the nine cells, 921 / 9 = 102.3 m: at any grade, cut minus fill is the sum of (ground minus grade) times the area, which is zero exactly at the mean.

Computed for the hypothetical pad on the Lab 5 DEM (tools/week10_volume_figures.py; 150 cells, ground 1,511.9 to 1,524.0 m, mean slope 8.1 %). Real balance is never exactly here: swell and shrink factors, side slopes and unusable material (topsoil, rock) all move it. Hauling is what costs money, which is why designers chase balance.

Spatial Analyst Cut Fill (arcpy.sa.CutFill), run in ArcGIS Pro 3.7.1 on the pad. The sign was checked with an off-balance run (grade raised 2 m): the tool's positive total, 9,487 m³, matched the cut computed cell by cell, and its negative total, -39,487 m³, matched the fill. VERIFY in the GUI before class: the Geoprocessing pane path (Spatial Analyst Tools > Surface > Cut Fill); it was not opened in the GUI for this deck.

The average-end-area method is the standard road earthwork estimate; the grid method is its raster cousin. Corridor design software computes sections from a DEM and a design profile, then sums them with this formula. Diagram drawn for this deck.

Earthwork contractors quote bank (in place), loose and compacted volumes, related by factors measured for the soil; the grid gives bank volume. No numbers for those factors are given here on purpose: they come from geotechnical tests for a site. The pad sits in a developed part of Provo's east bench, so the DEM's ground includes roads and lots; it is a hypothetical design for illustration.

A north-south profile through Big Southern Butte's summit from the Lab 9 DEM, with three flat bases taken from the outline's edge. Thursday shows that the choice of base moves the answer by about a fifth.

Four minutes, in pairs. If the room has no signal, put the URL on the board.

Week 10 due items from tools/build_schedule.py: Quiz 8 (Volumes and Spatial Analysis, Chapter 9) and Lab 9. Deck written October 9, 2026; figures from tools/week10_volume_figures.py, numbers in tools/week10_volume_numbers.json.