# What is the arc length?

Finds the arc length s = rθ of a circle from the radius and the central angle, or any two of the radius, angle, arc length, and sector area, with the chord.

- Page: https://www.acalculator.org/math/arc-length-calculator
- JSON spec: https://www.acalculator.org/math/arc-length-calculator.json
- Version: 21edb6e60a70

## Default answer

Example with the default inputs (Radius 10, Central angle 60 °): An arc with radius 10 and a central angle of 60° is 10.471976 long, and its sector has an area of 52.359878.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The radius r of the circle: from the centre to the arc. |
| angle | Central angle | The angle θ at the centre of the circle between the two radii that end the arc, up to 360°. |
| s | Arc length | The length s of the arc: the part of the circle between the two radii. |
| area | Sector area | The area A of the sector: the slice of the circle that the arc bounds. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| r | Radius | The radius r of the circle: from the centre to the arc. |
| theta | Central angle | The angle θ at the centre of the circle between the two radii that end the arc. |
| s | Arc length | The length s of the arc: the part of the circle between the two radii. |
| a | Sector area | The area A of the sector: the slice of the circle that the arc bounds. |
| chord | Chord length | The straight line between the two ends of the arc: 2r sin(θ/2). |
| perimeter | Sector perimeter | The distance around the sector: the arc plus two radii, s + 2r. |

## Method

s = rθ, where θ is the central angle in radians; the sector area is A = ½r²θ = ½sr.

## Assumptions

- The central angle is more than 0° and at most 360°. It can be typed in degrees or radians; the formulas use radians (θ in radians = degrees × π ÷ 180).
- The radius, arc length, and sector area are positive numbers in any one unit; the area is in that unit squared.
- An arc longer than the circumference, or a sector bigger than the circle, has no answer.

## Worked examples

1. r = 10, angle = 1.047198 gives s = 10.471976, area = 52.359878, chord = 10, perimeter = 30.471976. Source: OpenStax, Precalculus 2e, §5.1 Angles (https://openstax.org/books/precalculus-2e/pages/5-1-angles).
2. r = 5, s = 12 gives angle = 2.4, area = 30, chord = 9.320391.
3. s = 15, area = 60 gives r = 8, angle = 1.875, chord = 12.897298.
4. angle = 1.570796, area = 314.159265 gives r = 20, s = 31.415927, chord = 28.284271.
5. r = 3, angle = 6.283185 gives s = 18.849556, area = 28.274334, chord = 0.

## FAQ

### What is the arc length formula?

s = rθ, where r is the radius and θ is the central angle in radians. With the angle in degrees, s = 2πr × (θ ÷ 360°). An arc of 60° on a circle of radius 10 is 10 × π/3 = 10.472 long.

### How do I find the central angle from the arc length?

Divide the arc length by the radius: θ = s ÷ r, in radians. An arc of 12 on a circle of radius 5 has θ = 2.4 radians, which is 2.4 × 180° ÷ π = 137.51°.

### How do I find the radius from the arc length and the angle?

r = s ÷ θ, with θ in radians. An arc of 31.4159 with a central angle of 90° (π/2 radians) has a radius of 31.4159 ÷ 1.5708 = 20.

### What is the difference between arc length and chord length?

The arc follows the curve of the circle; the chord is the straight line between the two ends. The chord is always shorter: c = 2r sin(θ/2). For a 60° arc of radius 10, the arc is 10.472 and the chord is exactly 10.

### How is the sector area related to the arc length?

The sector is the slice of the circle bounded by the arc and two radii. Its area is A = ½r²θ, which is also ½ × arc length × radius. That is why the calculator can find the radius from the arc length and the sector area: r = 2A ÷ s.

### Why does the formula need radians?

A radian is defined so that an arc of one radius subtends an angle of 1 radian. That makes s = rθ exact with no conversion factor. The calculator accepts degrees and converts: radians = degrees × π ÷ 180.

### Can the central angle be more than 360°?

No. An arc on one circle can go at most once around, so the angle is at most 360° (2π radians) and the arc is at most the circumference 2πr. An arc longer than that has no answer here.

## Sources

- OpenStax, Precalculus 2e, §5.1 Angles (arc length s = rθ, area of a sector A = ½θr²). https://openstax.org/books/precalculus-2e/pages/5-1-angles
- NIST Digital Library of Mathematical Functions, §4.14 (sine) and §3.12 Mathematical Constants (π). https://dlmf.nist.gov/3.12