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How to use
You can derive the main dimensions of a circle from any one of four known measurements: radius, diameter, circumference or area. The tool first converts the input to a radius, then derives the other measurements using the same geometric relationships. Radius is half the diameter, circumference is 2π times radius, and area is π times radius squared. If a circular garden has a known boundary length, choose circumference rather than guessing its radius. If only floor area is known, select area and the tool recovers the radius using a square root. This is particularly useful when a design drawing, physical object or material estimate supplies a different known measurement.
- Identify which one of radius, diameter, circumference or enclosed area is actually provided by the problem.
- Enter a positive value with a consistent unit; lengths use units such as cm while areas must use matching squared units.
- Calculate all four outputs and read them separately: radius and diameter are lengths, circumference is boundary length, and area is squared.
- Check an answer by switching the known measurement to a derived value, allowing a small difference from rounding π.
How it is calculated
Example
For a circle with radius 7 cm, diameter is 14 cm. The circumference is 2π × 7 = 14π, approximately 43.98 cm, while the enclosed area is π × 7² = 49π, approximately 153.94 cm². Reverse the problem: if the measured circumference is 14π cm, radius is circumference divided by 2π, or 7 cm. If the given area is instead 49π cm², radius is the square root of area divided by π, again 7 cm. The two answers have different units: perimeter is linear, but area is measured in square centimetres.
Important notes
These formulas assume an ideal flat circle rather than an irregular boundary. Do not substitute diameter for radius in πr²: that makes the computed area four times too large. Calculated decimals approximate π, so reversing a rounded measurement can introduce small discrepancies. Keep units consistent: centimetres for radius and circumference imply square centimetres for area; metres imply square metres. This calculator covers the entire circle, not a partial circular sector. A sector's area or arc length additionally requires an angle and should be handled with the separate sector tool. Real-world measurement uncertainty is independent of numerical rounding.
Worked examples and interpreting results
A circle with radius 3 cm has diameter 6 cm, circumference 6π cm and area 9π cm². Radius and diameter are different inputs; using diameter as radius doubles circumference and quadruples area.
| Case | Calculation | Result |
|---|---|---|
| Diameter | 2 × 3 | 6 cm |
| Circumference | 2 × π × 3 | 18.849556 cm |
| Area | π × 3² | 28.274334 cm² |
| Diameter 12 cm to radius | 12 ÷ 2 | 6 cm |
| Area of a circle with 12 cm diameter | π × (12 ÷ 2)² | 113.097336 cm² |
How to check the result
Circumference uses a length unit; area uses a squared unit. Doubling the radius quadruples the area. Decimal results are rounded because π has no finite decimal expansion.
Circle calculator: area, circumference and radius formulas
For a circle of radius r, multiply π by r squared to obtain area, or multiply 2π by r to obtain circumference. If you only know diameter, halve it to get the radius first. The area uses squared length units whereas circumference uses ordinary length units.
- Area of a circle with radius 5: π × 5² = ≈ 78.5398
- Circumference of a circle with radius 5: 2π × 5 = ≈ 31.4159
- Area of a circle with radius 3: π × 3² = ≈ 28.2743
Before calculating
Do not substitute the full diameter for the radius in πr². Confirm whether the given measurement is radius or diameter, and remember that rounding π early can make small differences in the result.
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Frequently asked questions
How do I find radius from circumference?
Divide circumference by 2π. A circle with circumference 14π centimetres has radius 7 centimetres.
How do I calculate radius from area?
Take the square root of area divided by π. The area must use a unit squared consistently with the radius.
Does this calculate the area of a sector?
No. This page gives measurements of an entire circle; a sector requires its central angle too.