Cone Calculator
Calculate the volume and surface area of a cone from its radius and height.
About this calculator
A cone's volume comes out to exactly one-third of a cylinder sharing the same base radius and height: V = (1/3)πr²h. That factor of one-third is not a rounding convenience — it falls out of integrating the area of an infinite stack of shrinking circular disks from the apex down to the base, and it holds for any cone regardless of how tall or wide it is. Surface area is a separate calculation from volume: this calculator first derives the slant height (the straight-line distance from the apex down to the base's edge, found via the Pythagorean theorem from radius and height) and then adds the curved lateral surface, πr times the slant height, to the flat circular base, πr².
The result is the cone's total surface area, matching a party hat's paper if it also had a bottom disk, or a scoop of ice cream's cone including the flat rim at the top. If you only need the paper for the curved wrap itself with no base — the actual party-hat case — subtract πr² from the reported surface area. Because both height and slant height are measured as the perpendicular and diagonal distances from apex to base respectively, mixing them up is the most common input error: a cone's slant length is always longer than its vertical height for any real cone with positive radius, so if you have only the slant length on hand, convert it to true height first using height = √(slant² − radius²).
Inputs
Results
Volume
37.6991
How to Use This Calculator
- Enter the cone's base radius — the distance from the center of the circular base to its edge.
- Enter the cone's height — the perpendicular distance from the base to the apex, not the slanted side.
- Read the volume, which tells you how much the cone would hold or how much material fills it.
- Read the surface area, which covers the curved side plus the flat circular base — useful for a total material estimate.
How the result changes with Radius
| Radius | Volume |
|---|---|
| 1.5 | 9.4248 |
| 2.25 | 21.2058 |
| 4.5 | 84.823 |
| 7.5 | 235.6194 |
Engine last updated . Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
What is the difference between the cone's height and its slant height?
Height is the straight vertical distance from the apex down to the center of the base, measured perpendicular to the base. Slant height is the diagonal distance from the apex down to the edge of the base along the cone's outer surface, and it's always longer than the height for any cone with a nonzero radius. This calculator asks for height and derives slant height internally using the Pythagorean theorem.
Does the surface area include the circular base or just the curved side?
It includes both — the reported surface area is the curved lateral surface plus the flat circular base, giving the total outer skin of a solid cone. If you're estimating material for something open at the bottom, like a party hat or a filter cone with no floor, subtract the base area (πr²) from the total to get just the curved portion.
Why is a cone's volume exactly one-third of a cylinder with the same base and height?
It's a consequence of how a cone's cross-sectional area shrinks as you move from base to apex — at any height fraction along the cone, the cross-section is a circle scaled down, and integrating that shrinking area over the full height always yields exactly one-third of the constant cross-section a cylinder would have. This is a fixed geometric fact, true for every cone regardless of its specific proportions.
How much does doubling the height change the volume compared to doubling the radius?
Volume scales linearly with height but with the square of radius, since the formula is (1/3)πr²h. Doubling the height alone doubles the volume, but doubling the radius alone quadruples it — so for a cone-shaped stockpile or funnel, getting the radius right matters far more to the total volume than getting the height right.
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