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Circle Calculator: Radius, Diameter, Circumference and Area
Start from whichever measurement you know and get the rest. Add an optional central angle to also find the arc length, sector area, chord length and circular segment area.
How the four measurements are linked
Every circle is fixed by a single number, which is why you only need to enter one of the four. The radius r is the distance from the centre to the edge. The other three follow from it:
- Diameter:
d = 2r - Circumference:
C = 2πr = πd - Area:
A = πr²
Going the other way, the radius is d ÷ 2, or C ÷ 2π, or √(A ÷ π). The constant π (about 3.14159265) is the ratio of the circumference to the diameter of any circle, and the calculator uses the full-precision value stored by the browser rather than 3.14.
Arc, sector, chord and segment
A central angle cuts a wedge from the circle. With the angle θ in radians (degrees × π ÷ 180), the formulas are:
- Arc length:
r × θ, the part of the circumference inside the wedge. - Sector area:
½ r² θ, the wedge itself, like a slice of pizza. - Chord length:
2 r sin(θ ÷ 2), the straight line joining the two ends of the arc. - Segment area:
½ r² (θ − sin θ), the region between the chord and the arc.
For a 90 degree angle on a circle of radius 2, the arc is π (about 3.14), the sector is also π, the chord is 2.83 and the segment is about 1.14. A 180 degree angle gives a half circle whose chord is the diameter.
How size scales
Doubling the radius doubles the diameter and circumference but multiplies the area by four. Tripling the radius multiplies the area by nine. This square law explains why a 12 inch pizza holds far more than twice the food of a 6 inch one: its area is four times as large. The same effect applies to pipes. Doubling the inner diameter of a pipe lets about four times the cross-sectional area carry flow, although friction and other effects mean flow rate does not scale so neatly.
Common mistakes
- Mixing up radius and diameter. Measure edge to edge through the centre for the diameter, then halve it. Using the diameter in
πr²overstates the area by a factor of four. - Forgetting the units of area. If you enter a radius in centimetres, the area is in square centimetres, and the circumference is in centimetres.
- Rounding π too early. Using 3.14 gives an error of about 0.05%, which can matter in large structures.
- Entering degrees where radians are expected. This page takes degrees and converts internally.
See also the area calculator for other shapes and the midpoint calculator, since the centre of a circle is the midpoint of any diameter.
Worked example
A circular garden bed has an area of 100 square metres. What are its radius and circumference, and how long is the edge for a 60 degree wedge?
Inputs
| I know the | Area |
|---|---|
| Value | 100 |
| Central angle (optional) | 60 degrees |
Result
Radius 5.6419, diameter 11.2838.
| Radius r | 5.6419 |
|---|---|
| Diameter d = 2r | 11.2838 |
| Circumference C = 2πr | 35.4491 |
| Area A = πr² | 100 |
| Arc length for 60° | 5.9082 |
| Sector area | 16.6667 |
| Chord length | 5.6419 |
| Segment area (between chord and arc) | 2.8834 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I find the area of a circle from its circumference?
Divide the circumference by 2π to get the radius, then use area equals π times the radius squared. A circumference of 31.416 gives a radius of 5 and an area of about 78.54.
What is the relationship between radius and diameter?
The diameter is exactly twice the radius. Equivalently, the radius is half the diameter, measured from the centre to the edge rather than across the whole circle.
How do I find the arc length for an angle?
Multiply the circumference by the angle divided by 360. Equivalently, multiply the radius by the angle in radians. Both give the same length along the curve.
What is the difference between a sector and a segment?
A sector is the wedge bounded by two radii and an arc, while a segment is the region cut off by a chord, bounded by the chord and its arc.
Why is the circumference divided by diameter always pi?
All circles are scaled copies of one another, so the ratio of the distance around to the distance across is the same constant for every circle. That constant is called pi.
Related calculators
References and sources
- Standard Euclidean geometry: circle measurements, arc length and circular segment formulas.