Quick answer: a truncated cone (frustum) holds V = (1/3) π h (R² + Rr + r²), where R and r are the bottom and top radii. Don't average the radii and treat it as a cylinder — that underestimates, sometimes badly, because volume scales with radius squared.
The frustum formula, step by step
where R = larger radius · r = smaller radius · h = vertical height between the two circles
Worked example — a plant pot 12 in across the top, 8 in across the bottom, 10 in deep:
- Radii: R = 6 in, r = 4 in
- The bracket: 6² + (6 × 4) + 4² = 36 + 24 + 16 = 76
- V = (1/3) × π × 10 × 76 = 795.9 in³ ≈ 3.4 US gallons ≈ 13.0 liters
The three-term bracket is what makes this exact. Treating the pot as a cylinder with the average 5-inch radius would give π × 25 × 10 ≈ 785.4 in³ — about 1.3% low here, and the gap widens sharply as the taper steepens.
Two useful sanity checks are built into the formula: set r = 0 and it becomes the cone's (1/3)πR²h; set r = R and it becomes the cylinder's πR²h.
Common tapered containers — quick reference
| Container | Top ⌀ | Bottom ⌀ | Depth | Volume |
|---|---|---|---|---|
| 5-gallon bucket (to brim) | 11.9 in | 10.3 in | 13.5 in | ≈ 5.7 gal |
| 2-gallon bucket | 9.0 in | 7.5 in | 9.0 in | ≈ 2.1 gal |
| 10-inch plant pot | 10 in | 8 in | 9 in | ≈ 2.5 gal (9.4 L) |
| Paper coffee cup (12 oz) | 3.5 in | 2.2 in | 4.5 in | ≈ 13.4 oz to brim |
| Wheelbarrow drum (approx.) | — | — | — | 6 ft³ typical |
Every container in that list is sold by a nominal capacity slightly below its brim-full volume — the difference is deliberate headroom, so use inside measurements and expect the calculator to read a little high against the label.
Where frustum volume shows up in real life
Potting soil and container gardening
Nurseries size pots by nominal gallons, but the real fill volume comes from the frustum formula. A row of twelve 10-inch pots needs 12 × 9.4 ≈ 113 liters ≈ 4 ft³ of mix — roughly two large bags. The soil volume calculator converts to bag counts directly.
Hopper and silo transitions
Material bins taper from a wide barrel to a narrow discharge chute. A hopper going from 6 ft to 1 ft diameter over 4 ft holds (1/3)π × 4 × (9 + 1.5 + 0.25) ≈ 45 ft³. Getting this section right matters because it is the live storage that keeps a process running between refills.
Concrete pier footings
Bell-bottom and tapered piers are frustums. A pier 24 in at the base narrowing to 12 in over 18 in of depth takes (1/3)π × 18 × (144 + 72 + 36) ≈ 4,750 in³ ≈ 2.75 ft³ per pier — about 0.1 cubic yards each, which adds up fast across a deck.
Common mistakes & pro tips
- Averaging the radii. The most common shortcut, and it always reads low. Use the full three-term formula — that's what this calculator runs.
- Diameters entered as radii. Container widths are almost always quoted as diameters. Switch both fields to Diameter mode rather than halving by hand.
- Slant height as height. The h here is the vertical rise between the two circles. Convert a slant s with h = √(s² − (R − r)²).
- Outside dimensions on thick-walled pots. Terracotta and concrete containers have substantial walls — measure the inside for capacity.
- Pro tip — quick fill check. A tapered container filled halfway by depth is never half full by volume. On a typical bucket the bottom half holds only about 45% of the total.
Frequently asked questions
What is the formula for the volume of a truncated cone?
V = (1/3)πh(R² + Rr + r²), where R and r are the two radii and h the vertical height. A bucket 12 in across the top, 10 in across the bottom and 14 in deep holds (1/3)π × 14 × (36 + 30 + 25) ≈ 1,334 in³ ≈ 5.8 US gallons.
Why can't I just average the two radii?
Because volume depends on radius squared, not radius. Averaging the radii and treating the shape as a cylinder underestimates the true volume — for a bucket going from 6 in to 5 in radius the error is small, but for a steep taper like 10 in to 2 in it runs several percent low. The R² + Rr + r² term is the exact correction.
What is a frustum?
A frustum is any cone or pyramid with its top sliced off parallel to the base. A conical frustum — the shape on this page — is what most buckets, plant pots, drinking cups and lampshades actually are.
How do I find the volume of a bucket?
Measure the inside diameter at the top and at the bottom, halve each for the radii, and measure the inside depth. A standard 5-gallon bucket measures roughly 11.9 in top, 10.3 in bottom and 13.5 in deep inside, giving about 1,317 in³ ≈ 5.7 gallons to the brim — the nominal 5 gallons leaves headroom.
What happens if one radius is zero?
The formula collapses to (1/3)πR²h — the ordinary cone. And when both radii are equal it becomes πR²h, a cylinder. The frustum formula contains both of those as special cases, which is a useful way to sanity-check it.