Quick answer: a trapezoidal prism holds V = ½ (b₁ + b₂) × h × l. Average the two parallel widths, multiply by the perpendicular depth for the cross-section area, then multiply by the length. Sloped sides are handled exactly — no approximation involved.
The trapezoidal prism formula, step by step
where b₁, b₂ = the two parallel widths · h = perpendicular depth between them · l = length of the run
Worked example — a drainage ditch 4 ft wide at the top, 2 ft at the bottom, 1.5 ft deep, running 100 ft:
- Average width: (4 + 2) ÷ 2 = 3 ft
- Cross-section area: 3 × 1.5 = 4.5 ft²
- Times the run: 4.5 × 100 = 450 ft³ ≈ 16.7 cubic yards
Averaging the parallel sides is not a shortcut or an estimate — it is the exact area of a trapezoid, which is why this formula is the workhorse of earthwork estimating. Excavation contractors use it for every ditch, berm and roadway embankment they price.
Side slopes and how they change the numbers
Earthwork specs describe banks as a ratio — "3:1" means three horizontal for every one vertical. Given the bottom width, depth and slope ratio, the top width follows: b₁ = b₂ + 2 × (slope × depth).
| Side slope | Depth | Top width | ft³ per 100 ft | Cubic yards |
|---|---|---|---|---|
| 2:1 | 1 ft | 6 ft | 400 | 14.8 |
| 2:1 | 2 ft | 10 ft | 1,200 | 44.4 |
| 3:1 | 1 ft | 8 ft | 500 | 18.5 |
| 3:1 | 2 ft | 14 ft | 1,600 | 59.3 |
| 4:1 | 1 ft | 10 ft | 600 | 22.2 |
| 4:1 | 2 ft | 18 ft | 2,000 | 74.1 |
Notice how sharply the volume climbs with depth: doubling a 2:1 ditch from 1 ft to 2 ft triples the excavation. Flatter slopes are safer and easier to mow, but they move far more dirt.
Where trapezoidal prism volume shows up in real life
Drainage swales and roadside ditches
Stormwater design starts with capacity. A swale 6 ft at the top, 2 ft at the bottom, 1 ft deep, running 300 ft, holds ½ × 8 × 1 × 300 = 1,200 ft³ ≈ 8,977 gallons when full to the brim — the number that gets compared against the design storm runoff.
Retaining wall footings and berms
A trapezoidal footing 3 ft wide at the base, 2 ft at the top, 1 ft thick, running 40 ft: ½ × 5 × 1 × 40 = 100 ft³ ≈ 3.7 cubic yards of concrete. The concrete volume calculator adds bag counts and a waste allowance on top.
Tapered driveways and slabs
Slabs are often poured thicker at the edge that takes traffic. Treating the thickness taper as a trapezoid gets the pour right the first time — the difference between a 4-inch and a 6-inch average on a 400 ft² driveway is nearly 2.5 cubic yards, or most of a truckload.
Common mistakes & pro tips
- Sloped bank length used as depth. The h in this formula is the vertical distance between the parallel sides, measured perpendicular — never the diagonal face.
- Only one width measured. The whole point of the trapezoid is that the two parallel sides differ. If they're equal, it's just a box — use the rectangular prism calculator.
- Forgetting swell and compaction. Excavated soil expands 10–30% when loosened, and fill compacts 10–15% when placed. Bank volume from this calculator is the starting point, not the truck count — see the dirt volume calculator.
- Pro tip — irregular runs. Where a ditch changes cross-section along its length, break it into segments, compute each, and add. Surveyors call this the average end-area method, and it is exactly this formula applied piecewise.
Frequently asked questions
What is the formula for the volume of a trapezoidal prism?
V = ½ × (b₁ + b₂) × h × l — average the two parallel widths, multiply by the depth to get the trapezoid's area, then multiply by the prism's length. A ditch 4 ft across the top, 2 ft across the bottom, 1.5 ft deep and 100 ft long holds ½ × 6 × 1.5 × 100 = 450 ft³.
How do I calculate the volume of a ditch or swale?
A ditch is a trapezoidal prism lying on its side: b₁ is the top width, b₂ the bottom width, h the depth, and l the run. Averaging the two widths is what makes sloped sides work — the trapezoid area formula handles the taper exactly, with no approximation.
Which height do I use for a trapezoid?
The perpendicular distance between the two parallel sides — the true vertical depth, not the length of the sloped bank. On a 2:1 side slope, the sloped face is about 12% longer than the vertical depth, so using it inflates the volume noticeably.
Can I use this for a driveway or ramp on a slope?
Yes. A slab that is thicker at one end than the other has a trapezoidal side profile: b₁ and b₂ are the two thicknesses, h is the run, and l is the width. A driveway 6 inches thick at the garage and 4 inches at the street, 20 ft long and 10 ft wide, needs ½ × (0.5 + 0.333) × 20 × 10 ≈ 83.3 ft³ ≈ 3.1 cubic yards.