A cone holds exactly a third of the cylinder built around it — not half, which is the intuitive but wrong guess.
How it works
Squaring the radius, multiplying by pi and the height, and dividing by 3 gives the volume.
What this does not include
This does not include a frustum (a cone with the tip cut off), which needs a separate formula involving both the top and bottom radii.
How to use this calculator
- Choose whether you’re measuring by radius or diameter, then enter that value and the height.
A worked example
Radius 3, height 9: volume = (1/3)π × 3² × 9 = 84.823.
What the variables mean
| Variable | Meaning |
|---|---|
| Radius | Radius of the circular base |
| Height | Perpendicular distance from the base to the tip |
Edge cases worth knowing
A cone holds exactly a third of the cylinder built around it — same radius, same height — not half, a common intuitive but wrong guess.
The formula works for an oblique cone too (tip not centered over the base), since volume depends only on base area and the perpendicular height, not on where the tip sits.
Frequently asked questions
Why exactly a third, not a half?
It’s a direct result of integral calculus applied to how a cone’s cross-sectional area shrinks linearly from base to tip — the same one-third relationship holds for a pyramid inside its surrounding box.
What’s a practical use for this?
Estimating the volume of a conical pile (sand, gravel), a funnel, or an ice cream cone.
Does the cone need to be a “right” cone (tip directly above center)?
The volume formula works the same for an oblique cone too — volume depends only on base area and perpendicular height, not on whether the tip is centered.
