The Circle Area Calculator works out every property of a circle — area, radius, diameter and circumference — from whichever single value you already know. Most tools only go one way (radius in → area out), but this one solves in any direction: give it the circumference or the area and it will back-calculate the radius, then report all four quantities together. A live SVG diagram labels the radius and diameter, and a collapsible “Show working” section displays every step of the algebra so you can follow the calculation or copy it into a textbook answer.
It is entirely client-side — no numbers are sent to a server.
How the maths works
All four circle properties are locked together through the single constant π (pi ≈ 3.14159…). The relationships are:
| Starting from | Formula used to get radius (r) |
|---|---|
| radius r | r (no conversion needed) |
| diameter d | r = d / 2 |
| circumference C | r = C / (2π) |
| area A | r = √(A / π) |
Once the calculator has r, it derives the remaining three:
- Area = π × r²
- Circumference = 2 × π × r
- Diameter = 2 × r
The square-root step for the area-to-radius direction comes from rearranging A = π r² → r² = A / π → r = √(A / π). Similarly, C = 2πr rearranges to r = C / (2π). Every step is shown when you expand “Show working”.
Worked example
A circular swimming pool has a circumference of 25.13 m. What is its area?
- Find r from C: r = 25.13 / (2π) = 25.13 / 6.2832 ≈ 4.00 m
- Compute area: A = π × 4² = π × 16 ≈ 50.27 m²
So the pool surface covers just over 50 square metres. Select “Circumference” as the known value, type 25.13, choose “m” as the unit, and the calculator shows exactly that result with the full working below.
Formula reference table
| Quantity | Symbol | Formula |
|---|---|---|
| Area | A | π × r² |
| Circumference | C | 2 × π × r |
| Diameter | d | 2 × r |
| Radius from diameter | r | d / 2 |
| Radius from circumference | r | C / (2π) |
| Radius from area | r | √(A / π) |
The value of π used internally is Math.PI = 3.14159265358979323846…, accurate to
full double-precision floating-point. Results are rounded only for display, according
to the decimal-places selector (default 6).
Practical uses
Construction and landscaping. Circular patios, ponds, manhole covers and round windows all need area for material estimates and circumference for edging lengths.
Education. The tool shows each algebraic rearrangement step-by-step, making it useful for checking homework or demonstrating how to isolate a variable.
Engineering and design. Pipe cross-sections, wheel flanges and circular gaskets are sized by radius or diameter; knowing the exact cross-sectional area is needed for flow-rate and stress calculations.
Fabric and craft. Circular tablecloths, cake boards and embroidery hoops are often sold by diameter; this converts that to the fabric area you actually need to cut.
A note on precision
The formula A = π r² is exact in theory, but real measurements carry uncertainty. A tape measure typically reads to the nearest millimetre (±0.5 mm), which translates to a relative error in area of roughly 2 × (0.5 mm / r). For a circle of radius 100 mm that is about 1%; for radius 10 mm it is about 10%. Displaying 6 decimal places is mathematically correct but often false precision — choose 2 or 3 decimal places when reporting measured results.