⭕ Circle Area, Circumference & Diameter Calculator
Solve a circle from radius, diameter, circumference or area — plus arc length, sector, chord and segment, which the top results describe but never compute.
At a central angle of 90°
Results are unitless — whatever unit you enter, lengths come back in the same unit and areas in that unit squared. Enter millimeters and the area is in mm².
What Circle Area, Circumference & Diameter Calculator Does
A circle has only one free parameter. Fix the radius and everything else follows: the diameter is twice it, the circumference is 2πr, the area is πr². Which also means the reverse works — hand the calculator a circumference or an area and it can recover the radius and give you the rest.
That much every calculator for this term does, and Google now prints the formulas above the results anyway. What none of them computes is the more useful half of the subject: what happens when you only want part of the circle.
Cut a slice bounded by two radii and you have a sector. Draw a straight line between the two points where those radii meet the circle and you have a chord; the region between that chord and the arc is a segment. These are what you actually need for a pizza slice, a curved window, a circular tank filled partway, or a road bend of a given radius and angle.
The top-ranking page defines all of these terms carefully and then does not calculate any of them. This one does, for whatever central angle you give it, alongside the four standard values.
How to Use Circle Area, Circumference & Diameter Calculator
- Input the radius of the circle
- Review total computed area in square units
- Inspect diameter and circumference values
Formula Used by Circle Area, Circumference & Diameter Calculator
The four standard values
d = 2r · C = 2πr · A = πr²
- r
- Radius — center to edge
- π
- The ratio of circumference to diameter, about 3.14159 — irrational and transcendental
Worked example
A radius of 5
- d = 2 × 5 = 10
- C = 2 × π × 5 = 31.4159
- A = π × 5² = 78.5398
Result: And in reverse: r = C ÷ 2π = 5, or r = √(A ÷ π) = 5
Arc length and sector area
arc = rθ · sector = ½r²θ, with θ in radians
- θ
- Central angle in radians — degrees × π/180
- Fraction
- Both are simply that fraction of the whole: θ/2π of the circumference or area
Worked example
A 90° sector of a radius-5 circle
- 90° = π/2 radians
- arc = 5 × π/2 = 7.8540
- sector = ½ × 25 × π/2 = 19.6350
Result: Exactly a quarter of the circumference and a quarter of the area, as expected
Chord and circular segment
chord = 2r·sin(θ/2) · segment = ½r²(θ − sin θ)
- Chord
- The straight line between the arc's two endpoints
- Segment
- The sector with its triangle removed — the region between chord and arc
Worked example
The same 90° slice, radius 5
- chord = 2 × 5 × sin(45°) = 7.0711
- segment = ½ × 25 × (π/2 − sin 90°) = ½ × 25 × (1.5708 − 1)
Result: 7.1350 — far less than the 19.635 sector, because the triangle between the radii is most of it
One value gives you all four
The same circle described four ways. Enter whichever you have.
| If you know | Radius is | For r = 5 that value is |
|---|---|---|
| Radius r | r | 5 |
| Diameter d | d ÷ 2 | 10 |
| Circumference C | C ÷ 2π | 31.4159 |
| Area A | √(A ÷ π) | 78.5398 |
Parts of a circle at different angles
Radius 5 throughout. Note how the segment lags far behind the sector at small angles and catches up as the triangle shrinks.
| Angle | Arc length | Sector area | Chord | Segment area |
|---|---|---|---|---|
| 30° | 2.6180 | 6.5450 | 2.5882 | 0.2950 |
| 60° | 5.2360 | 13.0900 | 5.0000 | 2.2647 |
| 90° | 7.8540 | 19.6350 | 7.0711 | 7.1350 |
| 120° | 10.4720 | 26.1799 | 8.6603 | 15.3546 |
| 180° | 15.7080 | 39.2699 | 10.0000 | 39.2699 |
| 360° | 31.4159 | 78.5398 | 0.0000 | 78.5398 |
The vocabulary
Terms the top result defines but does not compute. Worth knowing which one your problem actually needs.
| Term | What it is |
|---|---|
| Chord | A straight line joining two points on the circle |
| Arc | Part of the circumference itself — minor if under half, major if over |
| Sector | The pizza slice between two radii and the arc |
| Segment | The region between a chord and its arc — the sector minus the triangle |
| Secant | A line cutting the circle at two points; a chord extended past both |
| Tangent | A line touching the circle at exactly one point, perpendicular to the radius there |
How to Read Your Result
Sector or segment — they are not the same thing
This is the distinction that sends people to a calculator in the first place. A sector includes the triangular wedge back to the center; a segment is only the sliver between the chord and the arc. At 90° on a radius-5 circle the sector is 19.635 and the segment is 7.135, a difference of nearly three to one, because the triangle accounts for most of the wedge. Getting the wrong one is a large error, not a rounding difference.
Why radians make the formulas simple
Arc length is rθ and sector area is ½r²θ only when θ is in radians. That is the whole reason radians exist — a radian is defined as the angle subtending an arc equal in length to the radius, so the arc formula becomes a plain multiplication with no conversion factor. In degrees you would carry a π/180 through every expression. Multiply degrees by π/180 to convert, or note that 180° is exactly π radians.
Area scales with the square of the radius
Doubling the radius doubles the circumference but quadruples the area. A 16-inch pizza has not twice the food of a 12-inch one but about 1.8 times, since area goes as r². This is also why a small change in a specified radius can matter far more than it looks for anything involving surface, volume or material cost, and why pipes are rated by area rather than diameter for flow.
The chord as a check on your answer
Two properties make the chord a useful sanity check. At 180° it equals the diameter exactly, because the line passes through the center. And for small angles it converges on the arc length, since a short arc is nearly straight. If your chord comes out longer than the diameter or longer than its own arc, something has gone wrong — probably degrees where radians were needed.
π is not just irrational
It is transcendental, which is a stronger statement: it is not the root of any polynomial with rational coefficients. Ferdinand von Lindemann proved this in 1882, and in doing so settled the ancient problem of squaring the circle — constructing a square of equal area with only compass and straightedge is impossible, because such a construction could only produce algebraic numbers. Two thousand years of attempts ended with a proof that the goal never existed.
Limitations & Accuracy Notes
- Results are unitless. Enter millimeters and lengths come back in millimeters with areas in mm²; the calculation makes no assumption about scale, but do not mix units within one problem.
- The central angle is capped at 360°. Angles beyond a full turn wrap around and describe the same geometry, so there is nothing to compute past that point.
- π is used to double-precision accuracy, so displayed values are correct to far more digits than any physical measurement justifies. The limiting factor is your input, not the arithmetic.
- This covers plane circles. Spheres, cylinders and arcs in three dimensions need different formulas, as does an ellipse — whose circumference has no elementary closed form at all.
- The pixel-grid circle generators that some searches want, for Minecraft builds and similar, are a different tool: they approximate a circle on a square lattice rather than computing its exact geometry.
Frequently Asked Questions
What are the circle formulas?
How do I find arc length and sector area?
What is the difference between a sector and a segment?
What is a chord?
Does the unit matter?
What is the relationship between radius, diameter and circumference?
Why does doubling the radius quadruple the area?
How precise is π here?
What is an arc length and a sector area?
Radians or degrees?
Is my data stored?
References & Further Reading
- NIST Digital Library of Mathematical Functions — Elementary geometry — US National Institute of Standards and Technology reference for the circular functions underlying the arc, chord and sector formulas
- Calculator.net — Circle Calculator — The most complete competing description of the parts of a circle, and the source of the transcendence-of-π history noted above