# What is the cylinder volume?

Finds the volume of a cylinder from its radius and height (V = πr²h), or the radius or height from the volume.

- Page: https://www.acalculator.org/math/cylinder-volume-calculator
- JSON spec: https://www.acalculator.org/math/cylinder-volume-calculator.json
- Version: 4ede7d3a7e52

## Default answer

Example with the default inputs (Radius 3 in, Height 5 in): A cylinder with radius 3 in and height 5 in has a volume of 141.4 in³.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The radius of the circular base: half its diameter. |
| h | Height | The distance between the two circular ends. |
| v | Volume | The space inside the cylinder. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The radius of the circular base: half its diameter. |
| h | Height | The distance between the two circular ends. |
| v | Volume | The space inside the cylinder. |

## Method

V = πr²h: the area of the circular base, πr², times the height h.

## Assumptions

- The cylinder is a right circular cylinder: two equal circles joined by a straight side. An oblique cylinder with the same base and vertical height has the same volume.
- 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 = 3, h = 5 gives v = 141.371669. Source: OpenStax Prealgebra 2e, Example 9.53 (141.3 with π ≈ 3.14); hand calculation in content.mdx: π × 3² × 5 = 45π = 141.3717.
2. r = 4, h = 13 gives v = 653.451272. Source: OpenStax Prealgebra 2e, Example 9.54 (653.12 with π ≈ 3.14); hand calculation in content.mdx: π × 4² × 13 = 208π = 653.4513.
3. v = 1,000, h = 10 gives r = 5.641896. Source: hand calculation in content.mdx: r = √(1000 ÷ (π × 10)) = 5.6419.
4. v = 500, r = 5 gives h = 6.366198. Source: hand calculation in content.mdx: h = 500 ÷ (π × 5²) = 6.3662.

## FAQ

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

V = πr²h. The base is a circle with area πr², and the cylinder is that circle stacked up to height h. A can with radius 3 in and height 5 in holds π × 9 × 5 = 141.37 in³.

### How many gallons does a cylindrical tank hold?

Work out the volume, then switch the volume unit to gallons. A tank 2 ft across (radius 1 ft) and 3 ft tall holds π × 1² × 3 = 9.42 ft³. One US gallon is 231 in³ and one cubic foot is 1,728 in³, so that is 9.42 × 1,728 ÷ 231 = 70.5 gallons.

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

Halve the diameter and type that as the radius. A pipe 10 cm across has a radius of 5 cm.

### How do I find the height from the volume?

Divide the volume by the base area: h = V ÷ (πr²). A 500 cm³ cylinder with radius 5 cm is 500 ÷ 78.54 = 6.37 cm tall.

### How do I find the radius from the volume?

Divide the volume by π times the height, then take the square root: r = √(V ÷ (πh)). A 1,000 cm³ (1 liter) cylinder 10 cm tall has a radius of 5.64 cm.

### Does the formula work for a slanted (oblique) cylinder?

Yes, if h is the vertical height between the two ends, not the length of the slanted side. Sliding the circular layers sideways does not change how much space they fill.

## Sources

- OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (V = πr²h; Examples 9.53 and 9.54). 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.
- NIST Handbook 44, Appendix C, General Tables of Units of Measurement: 1 US gallon = 231 in³. https://www.nist.gov/document/2026-nist-handbook-44-appendix-c