# What is the area of a circle?

Finds the area of a circle from its radius or diameter (A = πr²), or the radius and diameter from the area.

- Page: https://www.acalculator.org/math/area-of-a-circle-calculator
- JSON spec: https://www.acalculator.org/math/area-of-a-circle-calculator.json
- Version: 4e48a360265c

## Default answer

Example with the default inputs (Radius 5 in): A circle of radius 5 in (diameter 10 in) has an area of 78.54 in².

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The distance from the centre of the circle to its edge. |
| d | Diameter | The distance across the circle through its centre: twice the radius. |
| a | Area | The space inside the circle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The distance from the centre of the circle to its edge. |
| d | Diameter | The distance across the circle through its centre: twice the radius. |
| a | Area | The space inside the circle. |

## Method

A = πr², where r is the radius; with the diameter d = 2r, A = πd² ÷ 4.

## Assumptions

- π is the full-precision value 3.141592653589793…, not 3.14 or 22/7.
- The radius and diameter are in the length unit shown next to them, and the area is in the square unit shown next to it.
- Each value must be greater than 0.

## Worked examples

1. r = 5 gives a = 78.539816, d = 10. Source: hand calculation in content.mdx: A = π × 5² = 25π = 78.5398.
2. d = 14 gives a = 153.93804, r = 7. Source: hand calculation in content.mdx: r = 7, A = π × 7² = 49π = 153.9380.
3. a = 100 gives r = 5.641896, d = 11.283792. Source: hand calculation in content.mdx: r = √(100 ÷ π) = 5.6419.

## FAQ

### What is the formula for the area of a circle?

A = πr², where r is the radius. Square the radius first, then multiply by π. A radius of 5 cm gives 25 × 3.14159 = 78.54 cm².

### How do I find the area from the diameter?

Halve the diameter to get the radius, then use A = πr². In one step, A = πd² ÷ 4. A 14 in pizza has an area of π × 7² = 153.94 in².

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

Divide the area by π, then take the square root: r = √(A ÷ π). An area of 100 m² gives r = √31.831 = 5.64 m.

### Why does a 16 in pizza have more than twice the area of an 11 in pizza?

Area grows with the square of the size. A 16 in pizza has an area of π × 8² = 201 in², and an 11 in pizza has π × 5.5² = 95 in², so the larger one is about 2.1 times as much pizza.

### What units is the area in?

Square units: if the radius is in inches, the area is in square inches (in²). Each box has its own unit switch, so you can type the radius in feet and read the area in square metres.

### Is the area the same as πd?

No. πd is the circumference, the distance around the circle. The area is πr², the space inside it. They have different units: a length and a square length.

## Sources

- NIST Digital Library of Mathematical Functions, §3.12 Mathematical Constants, equation 3.12.1 (π = 3.14159 26535 89793…). https://dlmf.nist.gov/3.12
- OpenStax, Prealgebra 2e, §9.5 Solve Geometry Applications: Circles and Irregular Figures (A = πr²). https://openstax.org/books/prealgebra-2e/pages/9-5-solve-geometry-applications-circles-and-irregular-figures
- NIST Special Publication 811 (2008), Appendix B: 1 in = 0.0254 m exactly, for unit conversions.