frustum volume calculator
A frustum volume calculator uses V = πh ÷ 3 × (R² + Rr + r²). A cone cut flat with a 4 ft top, 8 ft base, and 6 ft height holds 175.93 ft³, or 6.516 yd³.
Frustum volume calculator: the formula
This frustum volume calculator finds the volume of a cone with its tip cut off flat using V = (π × h ÷ 3) × (R² + R × r + r²), where R is the bottom radius, r is the top radius, and h is the vertical height. Enter the top diameter, bottom diameter, and height in feet. The calculator halves the diameters for you and returns cubic feet, cubic yards (÷ 27), gallons (× 7.48052), and liters.
With the defaults, a 4 ft top, 8 ft bottom, and 6 ft height give 175.93 cubic feet.
Where the formula comes from
Picture the full cone before it was cut. The frustum is that big cone minus the small cone that was sliced off the top. Subtracting one cone formula from the other, and simplifying, gives the three-term bracket R² + Rr + r².
The bracket also explains why the answer is not just the average of the two ends. A frustum holds a little more than a cylinder built from the average radius, and the R × r term covers that difference.
Worked examples
- The default: 4 ft top, 8 ft bottom, 6 ft high
- R = 8 ÷ 2 = 4 ft, r = 4 ÷ 2 = 2 ft
- Bracket: 4² + 4 × 2 + 2² = 16 + 8 + 4 = 28
- π × h ÷ 3 = 3.1416 × 6 ÷ 3 = 6.2832
- Volume: 6.2832 × 28 = 175.93 ft³
- Cubic yards: 175.93 ÷ 27 = 6.516 yd³
- Gallons: 1,316.0 gal, or 4,981.8 L
- A tapered planter: 2 ft across the top, 1.5 ft across the bottom, 1.5 ft deep
- Volume: 3.63 ft³, or 27.2 gal
- Potting soil is usually sold in bags labeled by cubic feet or quarts. At 1.5 cubic feet per bag, that is 3.63 ÷ 1.5 = 2.4, so buy 3 bags.
- A flat-topped gravel pile: 20 ft across the base, 10 ft across the top, 8 ft high
- Volume: 1,466.08 ft³, or 54.299 yd³
Real piles slump unevenly, so treat this as an estimate. The stockpile volume calculator is built for heaps that come to a point.
Frustum vs cone vs cylinder
Same 8 ft base and 6 ft height, different tops:
| Top diameter | Shape | Volume |
|---|---|---|
| 0 ft | Cone | 100.53 ft³ |
| 4 ft | Frustum | 175.93 ft³ |
| 8 ft | Cylinder | 301.59 ft³ |
The frustum always falls between the cone and the cylinder, so if your answer is outside that range, an input is wrong.
Common frustum shapes
| Object | Typical use of the volume |
|---|---|
| Flowerpots and planters | Soil to fill them |
| Buckets and pails | Liquid capacity |
| Hoppers and funnels | Bin or chute capacity |
| Bell-bottom footings | Concrete for a flared pier base |
| Tapered columns | Concrete or fill |
| Round ponds with sloped sides | Water volume |
How to measure
- Diameters: measure straight across the widest part of each end, inside edge to inside edge if you want the capacity rather than the outside size.
- Height: measure vertically between the two flat faces. Lay a straightedge across the top and measure down to the bottom.
- Units: the boxes take feet. Convert inches by dividing by 12, so an 18 inch pot is 1.5 ft.
- Circumference only? Divide it by 3.1416 to get the diameter.
Common mistakes
- Entering radius instead of diameter. That cuts the volume to a quarter.
- Using the slant height. The sloped side is longer than the true height. On the default shape, the slant is about 6.32 ft against a true height of 6 ft.
- Averaging the diameters. Using a cylinder with a 6 ft average diameter gives 169.65 ft³ here, about 4% low. The error grows when the ends differ a lot.
- Forgetting the drainage layer. In planters, a gravel layer at the bottom reduces the soil needed.
For a plain cone, use the cone volume calculator. For a square-sided shape, see the pyramid volume calculator.
Frequently asked questions
What is a frustum, in plain terms?
A cone with its point sliced off by a cut parallel to the base. What remains has a large round bottom, a smaller round top, and sloping sides, like a bucket, a lampshade, or a flowerpot. This calculator handles the round (conical) frustum; a pyramid with its top cut off is a square frustum and uses a different area term.
What is the formula for the volume of a frustum?
V = (π × h ÷ 3) × (R² + R × r + r²), where R is the bottom radius, r is the top radius, and h is the vertical height. For a 4 ft top, 8 ft bottom, and 6 ft height: R = 4, r = 2, so V = (3.1416 × 6 ÷ 3) × (16 + 8 + 4) = 6.2832 × 28 = 175.93 cubic feet.
What if the top diameter is zero?
Then the shape is a full cone and the formula reduces to π × R² × h ÷ 3. With a 0 ft top, 8 ft bottom, and 6 ft height, the result is 100.53 cubic feet, which matches the cone volume calculator.
What if the top and bottom diameters are equal?
Then the sides are vertical and the shape is a cylinder. The formula gives π × R² × h, and with 8 ft at both ends and 6 ft of height you get 301.59 cubic feet. Any frustum must land between the cone and the cylinder of the same base and height.
Does the height need to be vertical?
Yes. Use the straight-up distance between the two flat faces, not the length along the sloping side. The slant length is always longer, so using it overstates volume. If you only know the slant length, the vertical height is the square root of slant² − (R − r)².
Does it matter which diameter I enter as the top?
No. The formula treats both ends the same way, so a flowerpot that is wider at the top gives the same volume as one standing upside down. Just make sure both diameters are in feet.
Estimates only. Verify quantities with your supplier before purchasing.