Is Your Shape a Frustum?
If it's round, tapers evenly, and has a flat top and a flat bottom of different sizes, yes. Buckets, flower pots, paper cups, lampshades, and many hoppers are all frustums: a cone with its tip cut off parallel to the base. "Truncated cone" and "conical frustum" are two names for the same shape.
Two limiting cases are worth knowing, because the calculator handles both without any special setting:
- Top radius 0: no top face, so the shape is a full cone. The Cone Volume Calculator gives the same answer.
- Equal radii: no taper, so the shape is a cylinder.
For a straight-sided rectangular tank with no taper, use the Tank Volume Calculator instead. Square or rectangular tapered shapes, like a hopper with flat sides, are pyramid frustums and have their own formula further down.
The Formula and Why Averaging Fails
R and r are the two radii (it doesn't matter which is top or bottom, the formula gives the same result either way) and h is the vertical height between the faces, not the sloping side. Halve any diameters first.
The tempting shortcut is to average the radii and treat the shape as a cylinder. Try it on a plant pot with a 9 cm top radius, 5 cm bottom radius, and 14 cm inside height:
| Method | Working | Volume |
|---|---|---|
| Frustum formula | π × 14 × (81 + 45 + 25) ÷ 3 | 2,213.78 cm³ |
| Average radius (7 cm) as a cylinder | π × 49 × 14 | 2,155.1 cm³ |
The average comes out about 2.7% low. It always undershoots, because the wide end holds more than the narrow end gives up. On a gentle taper like this pot the gap is small, but it grows as the two radii move further apart.
More Conical Frustums
| Shape | R, r, h | R² + Rr + r² | Volume |
|---|---|---|---|
| Cone with the top cut off | 12 cm, 4 cm, 18 cm | 144 + 48 + 16 = 208 | about 3,920.7 cm³ |
| Tapered container, diameters 20 in and 12 in | 10 in, 6 in, 15 in | 100 + 60 + 36 = 196 | about 3,078.8 in³ |
| Tapered tank | 0.8 m, 0.5 m, 1.2 m | 0.64 + 0.4 + 0.25 = 1.29 | about 1.621 m³ (1,621 L) |
The container row shows the usual trap: the problem gives diameters, so 20 and 12 become radii of 10 and 6 before anything is squared.
Measuring a Real Bucket
Buckets taper so they can stack inside each other, which makes them frustums. The number molded on the side is often a nominal size, and it may count volume right to the brim. To find what yours really holds:
- Measure the inside diameter across the rim and across the base, and halve both.
- Measure the inside height straight down, not along the sloping wall.
- Enter all three in the same unit and read the liters or gallons result.
For the volume at a fill line below the rim, measure the diameter at that line and use the fill depth as the height.
Square and Rectangular Frustums
When the faces are squares or rectangles rather than circles, work with the face areas instead of radii:
A square frustum with a 10 cm bottom, 6 cm top, and 8 cm height has A1 = 100 and A2 = 36, so √(A1A2) = 60 and V = 8 × (100 + 36 + 60) ÷ 3 ≈ 522.7 cm³. If the top face shrinks to a point, A2 = 0 and the formula becomes the ordinary pyramid formula, handled by the Pyramid Volume Calculator.
Units
The result is in the cube of your input unit, with liters, US and imperial gallons, and cubic feet shown alongside. By hand: 1,000 cm³ = 1 liter, 1 m³ = 1,000 liters, 231 in³ = 1 US gallon, 1 ft³ ≈ 7.48 US gallons.
How the Numbers Are Checked
Setting the top radius to 0 turns ((R^2 + Rr + r^2) × pi × h) / 3 into pi × R^2 × h / 3, because the Rr and r^2 terms both vanish. That is the standard cone formula, reached without any special-case code. Surface area is worked out separately: the slant height comes from sqrt((R − r)^2 + h^2) and feeds the lateral area on its own, independent of the volume math.
Frequently Asked Questions
How do you calculate the volume of a frustum?
For a conical frustum: V = (πh / 3)(R² + Rr + r²), where R is the larger radius, r is the smaller radius, and h is vertical height. For a pyramid frustum: V = h(A1 + A2 + √(A1 × A2)) / 3, where A1 and A2 are the two parallel face areas.
Is a frustum the same as a truncated cone?
Yes, same shape, different names. "Truncated cone" is common in engineering and manufacturing; "conical frustum" is the formal geometric term. Both refer to a cone with the top cut off parallel to the base.
Why can't I just average the two radii and use the cylinder formula?
Averaging underestimates volume. The frustum formula uses R² + Rr + r² rather than ((R+r)/2)² because the taper creates more volume near the larger end. The difference is small for gradual tapers but grows significant when R and r differ substantially.
Can top radius and bottom radius be switched?
Yes. The formula is symmetric in R and r, swapping labels does not change the result.
References
- Conical Frustum, Wolfram MathWorld: formal mathematical properties: volume formula, lateral surface area, and why the Rr term is necessary.
- Pyramidal Frustum, Wolfram MathWorld: the area-based volume formula for square, rectangular, and polygonal frustums.
- Frustum, Math is Fun: practical explanation with diagrams showing the difference between a frustum and a full cone.