How to calculate conical frustum volume in liters
Use both inside radii and perpendicular height to calculate a truncated cone without overestimating it as a cylinder.
What the calculation represents
A conical frustum models a circular vessel with straight sloping sides and two parallel ends of different sizes. It requires the larger inside radius R, smaller inside radius r, and perpendicular height h—not the sloping side length.
- Reference formula
V = π × h ÷ 3 × (R² + R × r + r²)- Unit reference
- 1 L = 1,000 cm³ · 1 m³ = 1,000 L · 1 US gal = 231 in³ = 3.785411784 L
A practical workflow
- Confirm that cross-sections are circular, the sloping wall is straight, and the ends are parallel. Split a profile with slope changes into multiple frustums.
- Measure larger and smaller inside diameters and halve them for R and r. Measure h perpendicular to the ends, not along the wall.
- Keep R, r, and h in one unit and substitute them in V = πh(R² + Rr + r²)/3.
- Convert the cubic result to liters, then separately allow for a domed base, internal parts, fill limit, or contents already occupying space.
Worked example
With a 60 cm inside top diameter, 36 cm inside bottom diameter, and 45 cm height, R = 30 cm and r = 18 cm. The volume is 83.13 liters. Treating it as a 60 cm cylinder gives 127.23 liters, 53% above the correct frustum value.
Reading the answer correctly
The equation is a poor model when walls curve, cross-sections are not circular, the base is domed, or wall thickness changes materially.
A cylinder based on the larger end always overstates a narrowing frustum. The correct equation combines both end areas and the Rr term; averaging diameters and calculating a cylinder is not exact either.
Three checks built into the equation
Equal radii must return cylinder volume, while a zero smaller radius must return cone volume. A true frustum lies between those limits.
For an open vessel, label top and bottom radii for clarity even though R and r are symmetric in the equation.
V = πh(R² + Rr + r²) ÷ 3when R = r: V = πr²hwhen r = 0: V = πR²h ÷ 3liters = cm³ ÷ 1,000
Comparison at a 45 cm height
Every case has a 30 cm larger radius. The table shows how strongly the smaller radius changes capacity.
Swipe horizontally to see every column
| Smaller radius | Resulting shape | Volume |
|---|---|---|
| 0 cm | Cone | 42.41 L |
| 10 cm | Conical frustum | 61.26 L |
| 18 cm | Conical frustum | 83.13 L |
| 24 cm | Conical frustum | 103.48 L |
| 30 cm | Cylinder | 127.23 L |
Checks before relying on the number
- The inputs are radii rather than diameters and both come from inside measurements.
- Height is axial/perpendicular, not slant height.
- When R = r the result matches πr²h; when r = 0 it matches a full cone.
- The result is labeled an ideal estimate when the wall curves or the vessel is oval rather than circular.
Practical questions
Can I enter diameters directly?
Only after rewriting the equation consistently. The safer workflow is to halve each diameter and use R and r exactly as defined.
Which height should I measure?
Measure perpendicular distance between the circular ends. The sloping wall length is slant height and cannot replace h.
How do I handle a vessel with two wall slopes?
Mark the change in slope, measure the radius at that section, and add two frustum volumes with their respective heights.
Is this the same as calculating potting mix?
The geometry may be identical, but this guide applies to any frustum vessel. Potting-mix quantity also subtracts drainage, root ball, and watering rim.
Sources and references
These references support the units, method, or limitations explained in this guide.
- NIST SI units: volume
Metrology reference for volume, liters, and cubic meters.
Keep both inside diameters and perpendicular height with the result. If the profile changes slope, sum separately measured frustum sections.
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