Truncated Pyramid Volume Calculator
A pyramid with its tip cut off — hoppers, planters, plinths and tapered excavations. Enter both parallel faces and the height between them.
- ✓ Square or rectangular
- ✓ Formula shown
- ✓ Both face areas given
Truncated pyramid volume
—
Quick answer: a truncated pyramid holds V = (h/3)(A₁ + A₂ + √(A₁A₂)), where A₁ and A₂ are the two parallel face areas and h is the perpendicular height. For square faces of edge a and b that becomes V = (h/3)(a² + b² + ab). The middle term is a geometric mean — averaging the two ends gives the wrong answer.
The frustum formula, step by step
square faces: V = (h/3)(a² + b² + ab)
where A₁, A₂ = parallel face areas · h = perpendicular height
Worked example — a hopper with a 6 m square top, a 4 m square bottom and 9 m between them:
- Face areas: 6² = 36 m² and 4² = 16 m²
- Geometric mean: √(36 × 16) = √576 = 24
- Sum and scale: (9 ÷ 3) × (36 + 16 + 24) = 3 × 76 = 228 m³
Compare that with the tempting shortcut of averaging the ends: (36 + 16) ÷ 2 × 9 = 234 m³. That is 6 m³ too much — over 2.5% — and the gap widens as the taper gets steeper.
Why the geometric mean, not the average
A frustum's cross-section shrinks linearly in edge length as you climb, which means its area shrinks with the square. Averaging areas assumes the section changes at a constant rate, and it does not — it changes fastest near the wide end. The √(A₁A₂) term is exactly the correction for that curvature.
The error is small for gentle tapers and large for aggressive ones, which is why it survives unnoticed in so many spreadsheets. This table shows the penalty for a 1 m tall square frustum with a 6 m base:
| Top edge | Correct volume | Averaging gives | Over-estimate |
|---|---|---|---|
| 5 m (gentle) | 30.33 m³ | 30.50 m³ | +0.5% |
| 4 m | 25.33 m³ | 26.00 m³ | +2.6% |
| 3 m | 21.00 m³ | 22.50 m³ | +7.1% |
| 2 m | 17.33 m³ | 20.00 m³ | +15.4% |
| 0 m (full pyramid) | 12.00 m³ | 18.00 m³ | +50% |
At the limit — a full pyramid — averaging overstates the volume by half. That is the clearest demonstration that the shortcut is not a rounding difference but a structural error.
The two checks that prove it
| Set the top face to… | Formula becomes | Which is |
|---|---|---|
| A₂ = A₁ | (h/3)(3A₁) = A₁h | A prism — no taper at all |
| A₂ = 0 | (h/3)(A₁) | A full pyramid |
Any frustum volume must therefore land between one third and one full base-times-height. If yours doesn't, an input is wrong.
Where truncated pyramids show up
Hoppers and bins
Material hoppers taper to a discharge opening so contents flow under gravity. The stored volume is a frustum, and getting it right decides batch sizing — see the tank volume calculator for vessels combining a frustum bottom with a straight shell.
Excavations with battered sides
Any pit dug in loose ground is cut back at an angle to stop the walls collapsing, so the hole is a frustum, not a box. Estimating it as a box under-orders the backfill — feed the result into the dirt volume calculator to add swell and get truckloads.
Planters, plinths and formwork
Tapered concrete plinths and planters are frustums. Pour volume converts straight to bags or yards with the concrete volume calculator.
Embankments and stockpiles
A flat-topped spoil heap is an inverted frustum. Where the pile comes to a point instead, use the cone calculator for round heaps or the pyramid formula for square ones.
Common mistakes & pro tips
- Averaging the two faces. The single most common error, and it always overestimates. Use the geometric mean.
- Using the slant height. h is the vertical distance between the faces, not the sloping edge or the slant along a face.
- Mixing units between faces. Both faces and the height must share one unit before multiplying.
- Pro tip — sanity-check the bounds. The answer must sit between (1/3)A₁h and A₁h. Outside that range, something is wrong.
- Pro tip — irregular faces still work. The formula only needs the two areas. Any polygon works, so split an awkward face into rectangles and triangles and total it.
How to use this calculator
- Choose square or rectangular. A square frustum needs one dimension per face; a rectangular one needs length and width for each face.
- Measure both parallel faces. Enter the larger bottom face and the smaller top face — the order does not affect the answer.
- Measure the perpendicular height between the two faces, straight up the middle — not the sloping edge or the slant along a side face.
- Read the volume. The calculator substitutes into V = (h/3)(A₁ + A₂ + √(A₁A₂)) and shows both face areas alongside the result.
Frequently asked questions
What is the formula for the volume of a truncated pyramid?
V = (h/3)(A₁ + A₂ + √(A₁A₂)), where A₁ and A₂ are the areas of the two parallel faces and h is the perpendicular height between them. For a square frustum with base edges a and b this simplifies to V = (h/3)(a² + b² + ab).
Why can't I just average the two ends?
Averaging the areas overestimates the volume. The correct middle term is the geometric mean √(A₁A₂), not the arithmetic mean. For a frustum with a 6 m and a 4 m square base, averaging gives 234 m³ against the true 228 m³ — a 2.6% error that grows as the taper steepens.
What is the difference between a truncated pyramid and a truncated cone?
Only the shape of the faces. A truncated pyramid has flat polygonal faces; a truncated cone has circular ones. The volume formula is identical in structure — both use (h/3)(A₁ + A₂ + √(A₁A₂)) — because it depends on the face areas, not their outlines.
Does the formula still work if the top face is zero?
Yes, and it becomes the ordinary pyramid formula. Setting A₂ to zero leaves V = (h/3)A₁, which is one third of the base area times the height, exactly as expected for a full pyramid.
How do I calculate a tapered excavation or embankment?
Treat it as an inverted frustum. Measure the rectangle at the top of the hole and the rectangle at the bottom, plus the depth, and the same formula applies. This is the standard method for pits with battered sides, where vertical-wall estimates would under-order the backfill.