# What is the cone volume?

Finds the volume of a cone from its base radius and height (V = πr²h ÷ 3), or the radius or height from the volume.

- Page: https://www.acalculator.org/math/cone-volume-calculator
- JSON spec: https://www.acalculator.org/math/cone-volume-calculator.json
- Version: f31dfd323894

## Default answer

Example with the default inputs (Radius 2 in, Height 6 in): A cone with radius 2 in and height 6 in has a volume of 25.13 in³.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The radius of the circular base: half its diameter. |
| h | Height | The vertical height from the base to the tip, not the slanted side. |
| v | Volume | The space inside the cone. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The radius of the circular base: half its diameter. |
| h | Height | The vertical height from the base to the tip, not the slanted side. |
| v | Volume | The space inside the cone. |

## Method

V = πr²h ÷ 3: one third of the base area πr² times the height h.

## Assumptions

- The cone has a circular base. The height is the vertical distance from the base to the tip, so the formula holds for a slanted (oblique) cone too.
- π is the full-precision value 3.141592653589793….
- The radius and height are in the length units shown next to them, and the volume is in the volume unit shown next to it.
- Each value must be greater than 0.

## Worked examples

1. r = 2, h = 6 gives v = 25.132741. Source: OpenStax Prealgebra 2e, Example 9.55 (25.12 with π ≈ 3.14); hand calculation in content.mdx: π × 2² × 6 ÷ 3 = 8π = 25.1327.
2. r = 2.5, h = 8 gives v = 52.359878. Source: OpenStax Prealgebra 2e, Example 9.56, diameter 5 (52.33 with π ≈ 3.14); hand calculation in content.mdx: π × 2.5² × 8 ÷ 3 = 52.3599.
3. v = 100, h = 6 gives r = 3.989423. Source: hand calculation in content.mdx: r = √(3 × 100 ÷ (π × 6)) = √(50 ÷ π) = 3.9894.
4. v = 50, r = 2 gives h = 11.936621. Source: hand calculation in content.mdx: h = 3 × 50 ÷ (π × 2²) = 11.9366.

## FAQ

### What is the formula for the volume of a cone?

V = πr²h ÷ 3, or one third of πr²h. The base is a circle of area πr², and h is the height from the base to the tip. A cone with radius 2 in and height 6 in holds π × 4 × 6 ÷ 3 = 25.13 in³.

### Why is a cone one third of a cylinder?

A cone and a cylinder with the same base and the same height differ in how fast their cross-sections shrink. Adding up the shrinking circles (with calculus, or by filling a cone three times to fill the cylinder) gives exactly one third.

### Do I use the height or the slant height?

The height: the straight-up distance from the centre of the base to the tip. The slant height runs along the side and is longer. If you only know the slant height s, the height is √(s² − r²).

### How much does an ice cream cone hold?

A cone 5 cm across (radius 2.5 cm) and 10 cm deep holds π × 2.5² × 10 ÷ 3 = 65.4 cm³, about 65 mL, if it is filled level with the rim.

### How do I find the height of a cone from its volume?

Multiply the volume by 3 and divide by the base area: h = 3V ÷ (πr²). A 50 in³ cone with radius 2 in is 150 ÷ 12.566 = 11.94 in tall.

### I know the diameter, not the radius. What do I type?

Halve the diameter and type that as the radius. A cone 5 in across has a radius of 2.5 in.

## Sources

- OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (V = πr²h ÷ 3; Examples 9.55 and 9.56). https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area
- NIST Digital Library of Mathematical Functions, §3.12 Mathematical Constants, equation 3.12.1 (π). https://dlmf.nist.gov/3.12
- NIST Special Publication 811 (2008), Appendix B: 1 in = 0.0254 m exactly, for unit conversions.