What this calculator does
A cone's slant height, its radius and its vertical height form a right triangle: the radius and the height are the two legs, and the slant height is the hypotenuse running from the rim up to the apex. That means the three are linked by the Pythagorean theorem, and if you know any two of them you can find the third.
This calculator handles the direction that is easy to need but awkward to work out by hand: given the radius and the slant height (often the two measurements you can actually take with a tape, since the vertical height runs through the inside of the cone), it solves for the vertical height using height = √(slant height² − radius²).
The formula
The radius, height and slant height of a cone satisfy slant height² = radius² + height², the same relationship as the two legs and hypotenuse of a right triangle. Rearranging for height gives height = √(slant height² − radius²).
| Term | Meaning |
|---|---|
| Radius | The distance from the centre of the circular base to its edge. |
| Slant height | The distance from any point on the base rim, along the outside surface, up to the apex. |
| Height | The perpendicular distance from the centre of the base straight up to the apex. |
The inputs explained
| Field | What to enter |
|---|---|
| Base radius | The radius of the cone's circular base. |
| Slant height | The slant height, measured along the outside surface from the rim to the apex. |
When to use it
Measuring a physical cone
The slant height and radius of a real cone, such as a funnel, a paper cone or a roof section, are easy to measure directly with a tape. The vertical height usually is not, since it runs through empty space inside the shape.
Working from a manufacturer spec
Some cone-shaped parts are specified by base diameter and slant height rather than vertical height, so this converts the given figures into the height needed for a volume calculation.
Checking the classic 3-4-5 case
A radius of 3 and a slant height of 5 gives a height of exactly 4, the same 3-4-5 right triangle used to check any Pythagorean calculation.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What height does a given radius and slant height produce?
A fixed radius of 6, checked against a range of slant heights.
Height for a fixed slant height, across different radii
The classic 5-12-13 right triangle and nearby radii, all sharing a slant height of 13.
Questions
How do you find the height of a cone from its slant height?
Use height = √(slant height² − radius²). You need both the radius and the slant height; the slant height alone is not enough.
How do you find the slant height of a cone instead?
If you already know the radius and vertical height, use slant height = √(radius² + height²), the forward direction of the same Pythagorean relationship, available on the main cone calculator.
What happens if the slant height is not longer than the radius?
That is not a valid cone: the slant height, as the hypotenuse of the right triangle, must always be the longest of the three measurements. Check the two input values if you see this message.
Does this work for an oblique (leaning) cone?
No. The Pythagorean relationship only holds for a right circular cone, where the apex sits directly above the centre of the base. An oblique cone needs different geometry entirely.
For the forward direction, computing volume, slant height and surface area from a known radius and height, see the cone calculator.