Area of a Circle Formula
The area of a circle is the amount of two-dimensional space enclosed within its boundary. The standard formula is:
A = π × r²
Where:
- A is the area (in square units)
- r is the radius — the distance from the center to any point on the edge
- π (pi) ≈ 3.14159265…
Since the diameter d = 2r, you can also express the formula as:
A = π × (d / 2)²
Both forms are equivalent. The calculator accepts any of the three values and derives the other two.
How to Calculate the Area of a Circle
From the radius
- Square the radius: r²
- Multiply by π: A = π × r²
Example: A circle with radius 5 cm A = π × 5² = π × 25 ≈ 78.540 cm²
From the diameter
- Halve the diameter to get the radius: r = d / 2
- Apply the formula: A = π × r²
Example: A circle with diameter 10 cm → r = 5 cm → A = π × 25 ≈ 78.54 cm²
From the area (finding the radius)
Rearrange A = π × r² to isolate r:
r = √(A / π)
Example: A circle has area 100 cm² r = √(100 / π) = √31.83 ≈ 5.64 cm d = 2 × 5.642 ≈ 11.28 cm
Practical Examples
| Shape | Radius | Diameter | Area |
|---|---|---|---|
| US quarter coin | 12.13 mm | 24.26 mm | 462.57 mm² |
| 12-inch pizza | 15.24 cm | 30.48 cm | 729.66 cm² |
| Round table | 0.6 m | 1.2 m | 1.13 m² |
| Circular swimming pool | 4 m | 8 m | 50.27 m² |
| Roundabout | 15 m | 30 m | 706.86 m² |
The Role of π
π (pi) is an irrational number — its decimal expansion never ends and never repeats. It is approximately:
- 3.14 — accurate to 2 decimal places; sufficient for rough estimates
- 3.14159 — accurate to 5 decimal places; adequate for most engineering work
- 22/7 — a convenient fraction approximation, accurate to within 0.04%
For precise calculations, always use a calculator or software that stores π to full precision. This tool uses JavaScript’s Math.PI (approximately 3.141592653589793).
Area vs. Circumference
The area of a circle measures the space inside it, in square units. The circumference measures the length of its outer boundary, in linear units:
| Measurement | Formula | For r = 5 cm |
|---|---|---|
| Area | A = π × r² | ≈ 78.54 cm² |
| Circumference | C = 2 × π × r | ≈ 31.42 cm |
Note that doubling the radius quadruples the area (because r is squared), but only doubles the circumference.