How long is the chord of the circle?
Type the radius and the central angle to get the chord length, c = 2r sin(θ/2). Or type the radius and the distance from the centre, the radius and the chord, or the chord and the height of the arc. The chord calculator also gives the angle, the arc length and the area of the segment.
- Chord length
- 10
The chord is 10 long.
- Radius
- 10
- Central angle (°)
- 60
- Central angle (rad)
- 1.047197551
- Distance from the centre
- 8.660254038
- Height of the arc (sagitta)
- 1.339745962
- Arc length
- 10.47197551
- Segment area
- 9.058607371
Chord length: 10. The chord is 10 long.
How to calculate
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.
Example with the default inputs (I know Radius and angle, Radius 10, Central angle 60 °): The chord is 10 long.
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.
- 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
Each example is checked against the calculator on every build.
- I know Radius and angle, Radius 10, Central angle 60° gives Chord length 10, Distance from the centre 8.660254, Height of the arc (sagitta) 1.339746, Arc length 10.471976, Segment area 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)
- I know Radius and distance, Radius 5, Distance from the centre 3 gives Chord length 8, Central angle (°) 106.260205, Height of the arc (sagitta) 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)
- I know Chord and height, Chord length 12, Height of the arc 2 gives Radius 10, Central angle (°) 73.739795, Distance from the centre 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)
- I know Radius and chord, Radius 1, Chord length 2 gives Central angle (°) 180, Height of the arc (sagitta) 1, Arc length 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)
How it works
A chord joins two points on a circle of radius r. The central angle θ is the angle at the centre between the radii to its two ends. The two radii and the chord form a triangle, so by the law of cosines c² = 2r² − 2r² cos θ = 4r² sin²(θ/2):
- chord c = 2r sin(θ/2)
- distance from the centre to the middle of the chord d = r cos(θ/2) (so c²/4 + d² = r²)
- height of the arc (sagitta) h = r − d = 2r sin²(θ/4)
- arc length s = rθ, θ in radians
- segment area A = r²(θ − sin θ)/2, the sector area ½r²θ minus the triangle ½r² sin θ
Pick what you know:
- Radius and angle. θ is more than 0° and less than 360°. Degrees become radians by θ × π/180.
- Radius and distance. θ = 2 arccos(d/r). The distance is from 0 up to, but not including, the radius.
- Radius and chord. θ = 2 arcsin(c/(2r)), the angle of the minor arc (at most 180°). The chord is at most 2r.
- Chord and height. r = c²/(8h) + h/2 and θ = 2 atan2(c/2, r − h). A height above r means the arc is the major arc, θ above 180°, and the distance d is then negative (the centre is on the arc's side of the chord). The chord and height shown are the ones typed.
Rules
- Every length typed is from 10⁻⁵⁰ to 10⁵⁰; the distance from the centre may also be 0. All lengths are in one unit of your choice.
- A distance at or above the radius, or a chord longer than the diameter, has no answer.
- The maths runs in double precision. θ − sin θ is worked out from its Taylor series θ³/6 − θ⁵/120 + θ⁷/5040 − θ⁹/362880 when θ is at most 0.01 rad, so a thin segment keeps its digits, and r²(θ − sin θ)/2 is formed as r × (r × (θ − sin θ)/2). A segment area that is not a positive finite double is left out.
Output format. Every value to 10 significant figures; the angle in degrees and in radians.
Worked examples by hand
Radius 10, angle 60°. c = 2 × 10 × sin 30° = 10. d = 10 cos 30° = 8.660254, h = 10 − 8.660254 = 1.339746, s = 10 × π/3 = 10.471976, A = 100 × (π/3 − sin 60°)/2 = 50 × (1.0471976 − 0.8660254) = 9.058607.
Radius 5, distance 3. c = 2√(25 − 9) = 8. θ = 2 arccos(0.6) = 106.260205°, h = 5 − 3 = 2.
Chord 12, height 2. r = 144 ÷ 16 + 1 = 10, d = r − h = 8, θ = 2 atan2(6, 8) = 73.739795°.
Radius 1, chord 2. The chord is the diameter: θ = 180°, h = 1, s = π.
Other questions people ask
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.