Circle Calculator
Enter any one of a circle’s radius, diameter, circumference or area, and get all four — with the results shown as exact multiples of π and as decimals.
Calculator
- Any known value → all others
- Area and circumference
- Exact π multiples
- Ratio check
Result & calculation receipt
- Method
- Derive the radius from the given quantity, then apply the circle formulas
- Formula
C = 2πr, A = πr², d = 2r- Substitution
- r = 5 ⟹ A = π·5² = 78.539816, C = 2π·5 = 31.415927
- Steps
- Radius r = 5 (from radius = 5).
- Area A = π r² = 78.53981634 (square units).
- Circumference C = 2π r = 31.41592654 (linear units).
- Diameter d = 2r = 10.
- Verification
- A ÷ C = 2.5 should equal r/2 = 2.5. ✓
- Exact result
- A = 25π, C = 10π
- Decimal result
- radius: 5 · diameter: 10 · circumference: 31.41592654 · area: 78.53981634
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator first recovers the radius from whatever you provide: it halves a diameter, divides a circumference by 2π, or takes the square root of an area over π. From the radius it computes the circle’s area (πr²) and circumference (2πr) and the diameter (2r). A ratio check confirms the results are consistent.
Worked example
A radius of 5 gives an area of 25π ≈ 78.54 and a circumference of 10π ≈ 31.42. Starting instead from a diameter of 10 recovers the same radius of 5 and the same results. Going backwards from an area of about 78.54 returns a radius of 5, closing the loop.
Key insight
Every circle measurement flows from the radius, and π is the single constant that links length to area. Because area depends on r², doubling a circle’s radius quadruples its area — a fact that surprises people sizing pizzas and pipes alike.
Frequently asked questions
What are the circle formulas?
Circumference C = 2πr, area A = πr², and diameter d = 2r, where r is the radius. Every other quantity is found by first recovering the radius.
How do I find the area from the circumference?
Divide the circumference by 2π to get the radius, then square it and multiply by π. The calculator does both steps and shows the radius it used.
Why is the area sometimes shown as a multiple of π?
Keeping the answer as, say, 25π is exact, while 78.54 is rounded. The calculator gives both so you can use whichever your problem needs.
What units does the area use?
If the radius is in some unit, the circumference is in that unit and the area is in that unit squared. Length and area units are never mixed.
Why does doubling the radius quadruple the area?
Because area is proportional to r², not r. Multiplying the radius by 2 multiplies the area by 2² = 4.
Methodology & verification
The radius is recovered from the given quantity (diameter ÷ 2, circumference ÷ 2π, or √(area ÷ π)), then the circle formulas C = 2πr, A = πr² and d = 2r are applied. The value must be positive; other inputs return a structured error. Results are reported as exact multiples of π and as decimals, and a ratio check (A ÷ C = r/2) confirms internal consistency.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.