Transparent method
The formula
Worked example
Example inputs
How to interpret the result
A cone's volume is exactly a third of a cylinder sharing the same base and height, a clean relationship that's easy to use as a sanity check on its own. It's also easy to mistake the cone's slant height, the tilted length running from base edge to apex, for its vertical height, and the two are different numbers whenever the cone isn't perfectly flat.
Before you rely on it
What to check
Confirm which height was measured: the straight vertical height, or the slant length along the side. The formula needs the vertical one.
The common error
Where people go wrong with cone volume & surface area calculator
Feeding the slant height into a formula that expects vertical height. The slant is always the longer of the two on any real cone, so this mistake consistently overstates the volume.
Sensitivity evidence
How radius changes the volume
| Radius | Volume | Total surface area |
|---|---|---|
| 2.4 | 48.3 | 81.1 |
| 2.7 | 61.1 | 94.5 |
| 3worked example | 75.4 | 108.8 |
| 3.3 | 91.2 | 123.9 |
| 3.6 | 108.6 | 139.9 |
Every input, tested
Which input moves the volume most
| Input | Tested from | To | Volume at each end | Swing |
|---|---|---|---|---|
| Radius | 2.7 | 3.3 | 61.1 to 91.2 | 30.2 (40%) |
| Height | 7 | 9 | 66 to 84.8 | 18.8 (25%) |
Two variables at once
Volume by radius and height
| Radius \ Height | 6 | 8 | 10 |
|---|---|---|---|
| 2.4 | 36.2 | 48.3 | 60.3 |
| 2.7 | 45.8 | 61.1 | 76.3 |
| 3 | 56.5 | 75.4 | 94.2 |
| 3.3 | 68.4 | 91.2 | 114 |
| 3.6 | 81.4 | 108.6 | 135.7 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Radius | 3 |
| Height | 8 |
| Volume | 75.4 |
| Total surface area | 108.8 |
Where the inputs come from
Enter the radius of the circular base and the vertical height, measured from the base to the tip, not along the slanted side. The calculator finds the slant height with the Pythagorean theorem for the surface area. Use one unit for both. Volume is one third of the volume of the cylinder with the same base and height, a relationship worth remembering for checking answers.
When the result stops applying
Real piles are not perfect cones, and the angle of repose of the material determines the shape. The slant height must not be confused with the vertical height. A truncated cone, or frustum, has a different formula. Rounding errors are small but units must be consistent.
What to compare it with
Cut a cone to make a frustum on the cone frustum calculator, and see cylinder, sphere and pyramid pages for other solids. The circle area page gives the base, the Pythagorean theorem page finds the slant height, and the concrete and gravel calculators in the trades categories estimate material.
People also ask
Frequently asked questions
Why is a cone's volume one third of a cylinder's?
It is a geometric result: a cone fills exactly one third of the cylinder with the same base and height, as calculus and models show.
What is the slant height?
The distance from the tip to the edge of the base along the side, found from the radius and height with the Pythagorean theorem.
How do I estimate a gravel pile's volume?
Treat it as a cone using its base radius and height, and remember that real piles slump, so the result is approximate.