Chord from radius and angle
c = 2r sin(θ/2)
The same chord borders both a minor and major segment.
Circle Chord Calculator — Chord Length, Sagitta and Central Angle
Enter Two Known Values
Chord and Segment Results
Radius—
Chord length—
Sagitta—
Central angle—
Arc length—
Segment area—
Enter two different positive measurements, then calculate to see the formula path.
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Circle Chord and Sagitta Formulas
A chord joins two points on a circle. The sagitta is the perpendicular distance from the chord to the midpoint of its arc. These formulas use central angle θ in radians.
Sagitta from radius and angle
s = r(1 − cos(θ/2))
For a minor segment, an equivalent form is s = r − √(r² − c²/4).
Radius from chord and sagitta
r = c²/(8s) + s/2
This works for minor and major sagittas.
Angle, arc, and segment area
θ = 2 acos(1 − s/r) and L = rθ.
A = ½r²(θ − sin θ).
Assumption: The tool models a straight chord in a Euclidean circle and a non-degenerate arc with
0° < θ < 360°. “Sagitta” means the perpendicular height from the chord midpoint to the selected arc: 0 < s ≤ r for a minor segment and r ≤ s < 2r for a major segment.
How to Use the Circle Chord Calculator
- Choose minor for the shorter arc or major for the longer arc. A semicircle belongs to either selection.
- Select two different known measurements from radius, chord length, sagitta, and central angle.
- Enter positive values and choose the shared length unit. Choose degrees or radians if one input is an angle.
- Select Calculate chord. The result card shows the solved geometry, formula path, and whether the segment is minor, major, or a semicircle.
All length inputs must use the same selected unit. Segment area automatically uses the square of that unit.
Worked Circle Chord Examples
Radius 10 cm and chord 12 cm
θ = 2 asin(12/20) ≈ 73.740°.
s = 10 − √(10² − 6²) = 2 cm.
The corresponding major segment has θ ≈ 286.260° and s = 18 cm.
Chord 16 m and sagitta 4 m
r = 16²/(8 × 4) + 4/2 = 10 m.
θ = 2 acos(1 − 4/10) ≈ 106.260°.
Radius 10 ft and major angle 240°
c = 2(10)sin(120°) ≈ 17.321 ft.
s = 10(1 − cos 120°) = 15 ft.
L = 10(4π/3) ≈ 41.888 ft.
Common Chord Calculation Mistakes
Using the wrong arc
One chord has a shorter and a longer arc. Radius and chord alone do not choose between them, so select minor or major before calculating.
Using degrees in the sine formula
The standard formulas assume radians. The calculator converts degree inputs automatically.
Confusing sagitta and apothem
Sagitta runs from the chord to the arc. The center-to-chord distance for a minor segment is r − s.
Circle Chord Calculator FAQs
How do I calculate chord length from radius and angle?
Use c = 2r sin(θ/2), where θ is the central angle. Convert degrees to radians before using the sine formula.
How do I find sagitta from chord length and radius?
For the minor segment, use s = r − √(r² − c²/4). The calculator uses an equivalent numerically stable form.
Can chord length and sagitta determine the radius?
Yes. Use r = c²/(8s) + s/2. The chord and sagitta must both be positive.
What is the difference between a minor and major segment?
A minor segment has a central angle no greater than 180° and a sagitta no greater than the radius. A major segment uses the longer arc, with an angle at least 180° and sagitta at least equal to the radius.
What happens when the chord is a diameter?
The central angle is 180°, the chord length is twice the radius, and the sagitta equals the radius. The minor and major segments are both semicircles.
How is circular segment area calculated?
With θ in radians, segment area is A = ½r²(θ − sin θ). This formula works for both minor and major segments.
Are my chord measurements uploaded?
No. The calculation runs locally in your browser and does not send your measurements to a server.
Calculation Notes
Last reviewed: July 30, 2026 by the Starlight Tools editorial team.
Calculations retain unrounded JavaScript floating-point values until display. Decimal places affect the page and copied summary; the downloaded CSV retains the underlying numeric results. Values that overflow or collapse beyond reliable browser precision are rejected.