# How big is my sphere?

Finds the radius, diameter, circumference, surface area, and volume of a sphere from any one of them, with A = 4πr² and V = 4/3 πr³.

- Page: https://www.acalculator.org/math/sphere-calculator
- JSON spec: https://www.acalculator.org/math/sphere-calculator.json
- Version: 587420afabc3

## Default answer

Example with the default inputs (Radius 5 in): A sphere of radius 5 in has a surface area of 314.159265 in² and a volume of 8,580.246646 cm³.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The distance from the centre of the sphere to its surface. |
| d | Diameter | The distance across the sphere through its centre: 2r. |
| c | Circumference | The distance around the sphere at its widest (a great circle): 2πr. |
| a | Surface area | The area of the whole surface: 4πr². |
| v | Volume | The space inside the sphere: 4/3 πr³. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The distance from the centre of the sphere to its surface. |
| d | Diameter | The distance across the sphere through its centre: 2r. |
| c | Circumference | The distance around the sphere at its widest: 2πr. |
| a | Surface area | The area of the whole surface: 4πr². |
| v | Volume | The space inside the sphere: 4/3 πr³. |

## Method

d = 2r, C = 2πr, A = 4πr², and V = 4/3 πr³, where r is the radius.

## Assumptions

- π is the full-precision value 3.141592653589793…, not 3.14 or 22/7.
- The sphere is perfect: every point of the surface is the same distance r from the centre.
- Each box has its own unit; each value must be greater than 0.

## Worked examples

1. r = 5 gives d = 10, c = 31.415927, a = 314.159265, v = 523.598776. Source: OpenStax, Prealgebra 2e, §9.6, sphere volume and surface area (https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area).
2. d = 12 gives r = 6, a = 452.389342, v = 904.778684.
3. v = 1,000 gives r = 6.203505, a = 483.597586.
4. a = 100 gives r = 2.820948, v = 94.031597.
5. c = 40,075,016.686 gives r = 6,378,137.000067.

## FAQ

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

V = 4/3 πr³, where r is the radius. A ball of radius 5 cm holds 4/3 × π × 125 = 523.6 cm³. If you know the diameter d, the same formula is V = πd³ ÷ 6.

### What is the formula for the surface area of a sphere?

A = 4πr². The surface of a sphere is exactly four times the area of a circle with the same radius (a great circle). A sphere of radius 5 cm has a surface area of 100π = 314.16 cm².

### How do I find the radius of a sphere from its volume?

Reverse V = 4/3 πr³: r = ∛(3V ÷ 4π). A volume of 1,000 cm³ gives r = ∛(3,000 ÷ 12.566) = ∛238.73 = 6.2035 cm.

### How do I find the radius from the surface area?

Reverse A = 4πr²: r = √(A ÷ 4π). A surface area of 100 in² gives r = √(100 ÷ 12.566) = 2.8209 in.

### What is the circumference of a sphere?

It is the length of a great circle, the biggest circle you can draw on the sphere (like the equator on a globe): C = 2πr = πd. It is what a tape measure reads when you wrap it around a ball at its widest.

### Why does doubling the radius make the volume eight times as large?

The volume grows with the cube of the radius, and 2³ = 8. The surface area grows with the square, so it becomes 2² = 4 times as large. This is why big balls hold much more than their surface suggests.

### Can I use different units for each box?

Yes. Each box has its own unit switch. Lengths are converted through metres, areas through square metres, and volumes through cubic metres, so a radius in inches and a volume in litres give the same sphere.

## Sources

- OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (sphere: V = 4/3 πr³, S = 4πr²). 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 (π = 3.14159 26535 89793…). https://dlmf.nist.gov/3.12
- NIST Special Publication 811 (2008), Appendix B: 1 in = 0.0254 m exactly, for unit conversions.
- NGA, World Geodetic System 1984, NGA.STND.0036 (equatorial radius 6,378,137 m). https://earth-info.nga.mil/