What this calculator does
A cone tapers from a circular base to a single point, and its volume is exactly one third that of a cylinder sharing the same base and height: a relationship first proved by Eudoxus and one of the earliest results in solid geometry to go beyond simple boxes and prisms.
The slant height, the distance along the sloped surface from the base edge to the apex, is not the same as the vertical height and is easy to confuse with it. It comes from Pythagoras, treating the radius, the height and the slant as a right triangle.
The formula
The slant height l = √(r² + h²) feeds directly into the curved surface area, πrl, which is what a cone’s side would measure if unrolled flat into a circular sector. Total surface area adds the base circle, πr², on top of that. The apex half-angle is the angle between the cone’s axis and its sloped side, found from the arctangent of radius over height.
| Term | Meaning |
|---|---|
| r | The radius of the circular base. |
| h | The vertical height, apex to base centre. |
| l | Slant height: apex to base edge along the surface, equal to √(r² + h²). |
| Apex half-angle | The angle between the axis and the sloped side. |
The inputs explained
| Field | What to enter |
|---|---|
| Base radius | The radius of the base circle. |
| Height | The vertical height from the base to the apex, not the slant. |
When to use it
Volume of a conical pile, funnel or hopper
Gravel, sand and grain settle into roughly conical piles. Measuring the base radius and the height gives the volume directly: useful for estimating a delivered load or a stockpile without moving it.
Material for a party hat, funnel or lampshade
The curved surface area is what the flat pattern for a cone-shaped object needs to cover: a wrap-around panel with no top or bottom included.
Comparing a cone to the cylinder it fits inside
A cone sharing a cylinder’s base and height always holds exactly a third of its volume, a fixed ratio regardless of the actual dimensions, useful as a quick sanity check on either figure.
Roof or spire geometry
A conical roof’s slant height is the true rafter length along the surface: the vertical height alone would undersize the roofing material needed.
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.
How volume and slant height change with radius, height fixed at 12
A constant height while the base radius widens.
| Base radius | Volume | Slant height | Apex half-angle |
|---|---|---|---|
| 3 | 113.097 | 12.369 | 14.04° |
| 5 | 314.159 | 13.000 | 22.62° |
| 9 | 1,017.88 | 15.000 | 36.87° |
| 12 | 1,809.56 | 16.971 | 45.00° |
| 16 | 3,216.99 | 20.000 | 53.13° |
How volume changes with height, radius fixed at 5
A constant base radius while the height stretches or shrinks.
| Height | Volume | Slant height | Apex half-angle |
|---|---|---|---|
| 5 | 130.900 | 7.071 | 45.00° |
| 12 | 314.159 | 13.000 | 22.62° |
| 20 | 523.599 | 20.616 | 14.04° |
| 30 | 785.398 | 30.414 | 9.46° |
| 50 | 1,309.00 | 50.249 | 5.71° |
Questions
What is the difference between slant height and vertical height?
Vertical height runs straight up the centre from base to apex. Slant height runs along the sloped outer surface from the base edge to the apex. They are only equal for a cone with zero radius, which is not really a cone at all.
How do I find the volume of a cone?
Multiply the base area, πr², by the height, then divide by three. The one-third factor is what distinguishes a cone’s volume from a cylinder sharing the same base and height.
Why is a cone’s volume exactly a third of the equivalent cylinder?
It is a general result for any pyramid or cone: a shape that tapers linearly to a point holds exactly one third the volume of the prism or cylinder with the same base and height, provable with calculus or, as the Greeks did, by exhaustion.
How do I find the curved surface area without the base?
Use πrl, where l is the slant height, not the vertical height. This gives the area of the sloped surface alone, matching what a flat pattern for the cone’s side would need.
How do I get the slant height if I only know the radius and vertical height?
Apply Pythagoras: slant height equals the square root of the radius squared plus the height squared, since the radius, height and slant form a right triangle.
For the cylinder this shape tapers from, see the cylinder calculator. For a pyramid with a rectangular rather than circular base, use the pyramid calculator.