Excavation volume with sloped sides
The one formula that gets a pit with sloped walls exactly right, and the two shortcuts that do not.
Last updated 29 September 2026The formula
Volume = Depth ÷ 6 × (A_base + 4 × A_mid + A_top)
A_base is the area at the bottom, A_top the area at ground level and A_mid the area at half the depth, all in square feet. Depth in feet gives cubic feet; divide by 27 for cubic yards.
For a rectangular pit with a bottom of L × W and walls sloping at s horizontal per foot of depth, the three areas are:
- A_base = L × W
- A_mid = (L + s × Depth) × (W + s × Depth)
- A_top = (L + 2 × s × Depth) × (W + 2 × s × Depth)
A worked example
Suppose a pit whose bottom is 40 ft by 30 ft, 10 ft deep, with walls sloping at 1 (one foot out for every foot down) on all four sides. The dimensions are inputs to show the method.
| Step | Working | Result |
|---|---|---|
| Base area | 40 × 30 | 1,200 sq ft |
| Mid-height area | (40 + 1 × 10) × (30 + 1 × 10) = 50 × 40 | 2,000 sq ft |
| Top area | (40 + 2 × 1 × 10) × (30 + 2 × 1 × 10) = 60 × 50 | 3,000 sq ft |
| Volume | 10 ÷ 6 × (1,200 + 4 × 2,000 + 3,000) | 20,333 cu ft |
| Cubic yards | 20,333 ÷ 27 | 753.1 cu yd |
The shortcuts, and how far off they are
Three quick answers people use for the same pit:
| Shortcut | Working | Volume | Versus the exact figure |
|---|---|---|---|
| Average of the top and base, times the depth | (1,200 + 3,000) ÷ 2 × 10 | 21,000 cu ft (777.8 cu yd) | +3.3% |
| Mid-height area times the depth | 2,000 × 10 | 20,000 cu ft (740.7 cu yd) | −1.6% |
| Base area times the depth (ignores the slope) | 1,200 × 10 | 12,000 cu ft (444.4 cu yd) | −41.0% |
Averaging the ends is the common shortcut, and it overstates because the area grows with the square of the distance out from the base, not in a straight line. Ignoring the slope leaves out 309 cubic yards of a pit whose exact volume is 753.
What the slope does to the volume
The same bottom and depth at other slopes:
| Side slope | Top of the pit | Volume | Times the vertical-walled pit |
|---|---|---|---|
| Vertical | 40 × 30 ft | 444.4 cu yd | 1.00× |
| 0.5 | 50 × 40 ft | 586.4 cu yd | 1.32× |
| 1 | 60 × 50 ft | 753.1 cu yd | 1.69× |
| 1.5 | 70 × 60 ft | 944.4 cu yd | 2.13× |
Which slope a pit needs, or whether it needs shoring instead, depends on the soil and the depth and is a safety decision for the competent person on site. OSHA’s excavation standard, 29 CFR 1926 Subpart P, governs it; this page does not say which slope to use.
A round pit
For a circular pit the same idea gives a frustum of a cone: Volume = π × Depth ÷ 12 × (D₁² + D₁ × D₂ + D₂²), with D₁ the bottom diameter and D₂ the top. A bottom of 20 ft, 8 ft deep at a slope of 1, has a top of 36 ft, so π × 8 ÷ 12 × (20² + 20 × 36 + 36²) = 5,060 cu ft, or 187.4 cubic yards.
After the dig
Everything above is a volume in the ground, a bank volume. The material occupies more room once it is loose in a truck, which is what bank vs loose cubic yards covers, and the swell factor is yours to supply. For a site whose ground is not flat, the cut and fill method handles the surface a single pit formula cannot.
What this leaves out
- Pits whose sides slope at different ratios, or whose floor is not level: split them into parts, or use the grid method.
- Over-excavation and working space: measure the bottom you actually dig.
- Ramps, benches and access cuts, which are extra excavation.
- Any price. The calculators here take the rate from you.
- 29 CFR 1926 Subpart P — Excavations — Occupational Safety and Health Administration
- The prismoidal and frustum volumes are defined by the formulas above — Arithmetic — verifiable by hand; recomputed here from the excavation calculator’s own functions
Put it to work
All earthwork calculators →Enter your own bottom, depth and slope and get the exact volume, with the working shown.