Circle Calculator
Enter whichever measurement you have — radius, diameter, circumference or area — and get all the others.
The formulas
π is the ratio of any circle's circumference to its diameter, approximately 3.14159265. It is the same for every circle of every size, which is the fact that makes all of this work, and it is irrational — its decimal expansion never terminates or repeats.
For everyday purposes 3.1416 is ample. NASA uses 15 decimal places for interplanetary navigation; 40 would be enough to compute the circumference of the observable universe to within the width of a hydrogen atom.
Arcs, sectors and chords
A sector is a wedge of the circle and an arc is the curved edge of that wedge. Both are simple fractions of the whole:
A 90° sector is exactly a quarter of the circle. The chord — the straight line joining the two ends of the arc — is shorter than the arc, which is the whole reason a road following a curve is longer than the line across it.
In radians rather than degrees these simplify further: arc length = rθ and sector area = ½r²θ. That elegance is precisely why radians exist.
Practical applications
- Area scales with the square of the radius. A 16-inch pizza has 78 % more area than a 12-inch one, not 33 % more. Two 12-inch pizzas give less food than one 16-inch. This is the single most useful thing on this page.
- Pipes and cables. Flow capacity depends on cross-sectional area, so doubling the diameter quadruples the capacity.
- Wheels. Distance travelled per revolution equals the circumference, which is how bicycle computers and odometers are calibrated.
- Circular efficiency. For a given perimeter, a circle encloses the maximum possible area. This is why bubbles, planets and storage tanks are round.
Frequently asked questions
Why does doubling the radius quadruple the area?
Because the radius is squared in the formula. Area = πr², so doubling r multiplies the area by 2² = 4. The circumference, where r appears to the first power, merely doubles.
What is the difference between degrees and radians?
Two ways of measuring the same angle. A full turn is 360° or 2π radians, so one radian is about 57.3°. Radians are defined as arc length divided by radius, which makes formulas like arc = rθ come out clean — and is why calculus uses them exclusively.
How do I find the area of a ring?
Subtract the inner circle from the outer: π(R² − r²). For a pipe with a 10 cm outer radius and 8 cm inner radius, the wall cross-section is π(100 − 64) = 113.1 cm².