# Circle calculator: area, circumference and radius

> Area, circumference, radius and diameter of a circle from any one of them, plus arc length, chord, sector and segment area for a central angle.

Interactive version: https://www.calcopenly.com/geometry/circle-calculator
Subject: Geometry calculators

Give any one of the radius, diameter, circumference or area and the other three follow from d = 2r, C = 2πr and A = πr². For a central angle θ the calculator also returns the arc length s = rθ (θ in radians), the chord c = 2r sin(θ/2), the sector area πr² × θ/360° and the segment area r²/2 × (θ − sin θ).

The default radius of 5 cm gives a diameter of 10 cm, a circumference of 31.4159 cm and an area of 78.5398 cm². Its 60° arc is 5.236 cm long and the chord across it is exactly 5 cm, because two radii and that chord form an equilateral triangle. Starting from the circumference suits trees, pipes and columns whose diameter cannot be reached: a girth of 100 cm means a diameter of 31.831 cm.

Inputs can be in any length or area unit, including hectares and acres, and every result appears in the unit chosen under "Show results in". The central angle must be above 0° and at most 360°; at 360° the segment is the whole disc.

## Inputs

- **I know the** (options: Radius, Diameter, Circumference, Area)
- **Radius**
- **Diameter**
- **Circumference**
- **Area**
- **Central angle for arc and chord**: Between 0° and 360°
- **Show results in** (options: Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km), Inches (in), Feet (ft), Yards (yd), Miles (mi))

## Results

- Area — main result
- Radius
- Diameter
- Circumference
- Arc length
- Chord length
- Sector area
- Segment area

## Formula

$$
\begin{gathered} A = \pi r^2,\quad C = 2\pi r,\quad s = r\theta \\ c = 2r\sin\tfrac{\theta}{2},\quad A_\text{segment} = \tfrac{r^2}{2}(\theta - \sin\theta) \end{gathered}
$$

## Worked examples

### Radius 5 cm, 60° arc

- I know the: Radius
- Radius: 5 cm
- Central angle for arc and chord: 60 °
- Show results in: Centimetres (cm)
- **Area: 78.539816 cm²**
- **Circumference: 31.415927 cm**
- **Diameter: 10 cm**
- **Arc length: 5.235988 cm**
- **Chord length: 5 cm**
- **Sector area: 13.089969 cm²**
- Checked against: Python 3.8 math: pi*5**2, 2*pi*5, 5*radians(60), 2*5*sin(radians(30)) = 5 (equilateral triangle), pi*25*60/360

### Circumference 100 cm

- I know the: Circumference
- Circumference: 100 cm
- Central angle for arc and chord: 60 °
- Show results in: Centimetres (cm)
- **Radius: 15.915494 cm**
- **Area: 795.774715 cm²**
- **Diameter: 31.830989 cm**
- Checked against: Python 3.8 math: 100/(2*pi), 100**2/(4*pi)

### Area 1 m², results in metres

- I know the: Area
- Area: 1 m²
- Central angle for arc and chord: 60 °
- Show results in: Metres (m)
- **Radius: 0.56419 m**
- **Circumference: 3.544908 m**
- Checked against: Python 3.8 math: sqrt(1/pi), 2*sqrt(pi)

### Diameter 12 in, quarter arc

- I know the: Diameter
- Diameter: 12 in
- Central angle for arc and chord: 90 °
- Show results in: Inches (in)
- **Chord length: 8.485281 in**
- **Sector area: 28.274334 in²**
- **Segment area: 10.274334 in²**
- **Arc length: 9.424778 in**
- Checked against: Python 3.8 math: 12*sin(pi/4), 36*pi/4, 36*pi/4 − 18, 6*pi/2

### Full turn (edge case): chord 0, segment is the whole disc

- I know the: Radius
- Radius: 2 m
- Central angle for arc and chord: 360 °
- Show results in: Metres (m)
- **Chord length: 0 m**
- **Arc length: 12.566371 m**
- **Segment area: 12.566371 m²**
- **Area: 12.566371 m²**
- Checked against: Definition: a 360° arc is the whole circumference; Python 3.8 math: 4*pi

### Angle typed in radians (π/3)

- I know the: Radius
- Radius: 5 cm
- Central angle for arc and chord: pi/3
- Show results in: Centimetres (cm)
- **Chord length: 5 cm**
- **Arc length: 5.235988 cm**
- Checked against: A 60° chord equals the radius (equilateral triangle); Python 3.8 math: 5*pi/3

## Questions

### How do you find the area of a circle from the diameter?

Use A = πd²/4, which is πr² with the radius written as half the diameter. A 10 cm diameter gives 25π ≈ 78.54 cm², and a 12 in pizza covers 36π ≈ 113.10 in². Area grows with the square of the diameter, so one 16 in pizza (201.06 in²) has more area than two 11 in pizzas together (190.07 in²).

### How do you find the radius from the circumference?

Divide the circumference by 2π: r = C/(2π). A tree with a girth of 100 cm has a radius of 15.915 cm and a diameter of 31.831 cm (C/π). Foresters' diameter tapes do this division for you: their scale is graduated in units of π, so wrapping the tape round the trunk reads the diameter directly.

### What is the difference between a sector and a segment of a circle?

A sector is the pie slice between two radii and the arc; a segment is the region between the arc and the chord joining its ends. The segment is the sector minus the triangle formed by the two radii and the chord. For a 90° slice of a circle with a 6 in radius, the sector is 9π ≈ 28.274 in², the triangle 18 in², and the segment 10.274 in².

### How do you calculate arc length?

Multiply the radius by the central angle in radians: s = rθ. With the angle in degrees, use s = 2πr × θ/360. A 60° arc on a 5 cm radius is 5 × π/3 ≈ 5.236 cm, one sixth of the 31.416 cm circumference. Putting degrees straight into s = rθ gives an answer 57.3 times too large, the number of degrees in one radian.

### Is 3.14 accurate enough for pi?

For estimates, yes. 3.14 is 0.05% below π, so a 5 cm radius gives an area of 78.5 cm² instead of 78.54 cm²; the fraction 22/7 is 0.04% too high. The absolute error grows with size: for a radius of 100 m, 3.14 gives 31,400 m², which is 15.9 m² short of the true 31,415.9 m².

### How accurate is the circle calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “Radius 5 cm, 60° arc” is checked against Python 3.8 math: pi*5**2, 2*pi*5, 5*radians(60), 2*5*sin(radians(30)) = 5 (equilateral triangle), pi*25*60/360.

### Where does the method come from?

Weisstein, E. W. “Circle”, “Circular Segment” — MathWorld; NIST Digital Library of Mathematical Functions §3.12 — mathematical constant π.

## Sources

- [Weisstein, E. W. “Circle”, “Circular Segment” — MathWorld](https://mathworld.wolfram.com/CircularSegment.html)
- [NIST Digital Library of Mathematical Functions §3.12 — mathematical constant π](https://dlmf.nist.gov/3.12)