# How long is the chord of the circle?

Finds the length of a chord of a circle, c = 2r sin(θ/2), from the radius and the central angle, the distance from the centre, or the arc height, with the arc length and the segment area.

- Page: https://www.acalculator.org/math/chord-calculator
- JSON spec: https://www.acalculator.org/math/chord-calculator.json
- Version: 259aca941b09

## Default answer

Example with the default inputs (I know Radius and angle, Radius 10, Central angle 60 °): The chord is 10 long.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| know | I know | Which two measures of the circle you know. |
| r | Radius | The radius r of the circle. |
| angle | Central angle | The angle θ at the centre between the two radii to the ends of the chord, more than 0° and below 360°. |
| d | Distance from the centre | The distance d from the centre of the circle to the middle of the chord, from 0 up to the radius. |
| c | Chord length | The straight-line length c between the two ends of the chord. |
| h | Height of the arc | The sagitta h: the distance from the middle of the chord to the arc, on the side of the arc you mean. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| chord | Chord length | c = 2r sin(θ/2), in the unit of the lengths typed. |
| radius | Radius | The radius r, typed or found from the chord and height: r = c²/(8h) + h/2. |
| degrees | Central angle (°) | The central angle θ in degrees. |
| radians | Central angle (rad) | The central angle θ in radians. |
| distance | Distance from the centre | d = r cos(θ/2): from the centre to the middle of the chord; negative when the arc is the major arc. |
| height | Height of the arc (sagitta) | h = r − d = 2r sin²(θ/4): from the middle of the chord to the arc. |
| arc | Arc length | s = rθ: the length of the arc that the chord cuts off. |
| segment | Segment area | A = r²(θ − sin θ)/2: the area between the chord and the arc, in square units. |

## Method

c = 2r sin(θ/2); θ = 2 arccos(d/r) or 2 arcsin(c/2r); from chord and height, r = c²/(8h) + h/2 and θ = 2 atan2(c/2, r − h); d = r cos(θ/2), h = r − d, s = rθ, segment area r²(θ − sin θ)/2.

## Assumptions

- Lengths are plain numbers in any one unit; every length output is in that unit and the area in its square.
- From the radius and the chord, the angle is the one at most 180° (the minor arc).
- From the chord and height, a height above the radius means the arc is the major arc (more than 180°).
- The maths runs in double precision.

## Worked examples

1. know = angle, r = 10, angle = 1.047198 gives chord = 10, distance = 8.660254, height = 1.339746, arc = 10.471976, segment = 9.058607. Source: OpenStax, Algebra and Trigonometry 2e, §10.2 Non-right Triangles: Law of Cosines (a² = b² + c² − 2bc cos α), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines (retrieved 2026-10-02); OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (arc length s = rθ, sector area ½θr²), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (retrieved 2026-10-02).
2. know = distance, r = 5, d = 3 gives chord = 8, degrees = 106.260205, height = 2. Source: OpenStax, Algebra and Trigonometry 2e, §10.2 Non-right Triangles: Law of Cosines (a² = b² + c² − 2bc cos α), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines (retrieved 2026-10-02).
3. know = height, c = 12, h = 2 gives radius = 10, degrees = 73.739795, distance = 8. Source: OpenStax, Algebra and Trigonometry 2e, §10.2 Non-right Triangles: Law of Cosines (a² = b² + c² − 2bc cos α), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines (retrieved 2026-10-02).
4. know = chord, r = 1, c = 2 gives degrees = 180, height = 1, arc = 3.141593. Source: OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (arc length s = rθ, sector area ½θr²), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (retrieved 2026-10-02).

## FAQ

### What is a chord of a circle?

A chord is a straight line between two points on a circle. The longest chord passes through the centre: it is the diameter, 2r. Every other chord is shorter and cuts the circle into a smaller part (the minor segment) and a larger part (the major segment).

### How do I find the length of a chord?

With the radius r and the central angle θ, the chord is c = 2r sin(θ/2). With the radius and the distance d from the centre to the chord, it is c = 2√(r² − d²). A circle of radius 5 with a chord 3 from the centre has c = 2√(25 − 9) = 8.

### Where does c = 2r sin(θ/2) come from?

The two radii and the chord make a triangle with two sides r and the angle θ between them. The law of cosines gives c² = r² + r² − 2r² cos θ = 2r²(1 − cos θ) = 4r² sin²(θ/2), so c = 2r sin(θ/2).

### How do I find the radius from a chord and its height?

The height h of the arc above the chord is called the sagitta. Then r = c²/(8h) + h/2. A chord of 12 with an arc 2 high comes from a circle of radius 144/16 + 1 = 10. This is how you find the radius of an arch or a curved edge.

### What is the segment area?

The segment is the region between the chord and its arc. Its area is the sector area minus the triangle: A = r²(θ − sin θ)/2 with θ in radians. For r = 10 and θ = 60°, A = 50 × (π/3 − sin 60°) ≈ 9.059.

### What units does the chord calculator use?

Lengths are plain numbers in any one unit: type the radius in inches and every length comes out in inches, and the segment area in square inches. The angle can be typed in degrees or radians.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §10.2 Non-right Triangles: Law of Cosines (a² = b² + c² − 2bc cos α), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines (retrieved 2026-10-02)
- OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (arc length s = rθ; sector area ½θr²), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (retrieved 2026-10-02)