Math
Chord Length Calculator
The chord length calculator solves the chord of a circle three ways: from the radius and the central angle (c = 2r·sin(θ/2)), from the radius and the chord's perpendicular distance from the center (c = 2·sqrt(r² − d²)), or backward from a measured chord and sagitta to recover the radius.
Every mode also reports the arc length, sagitta (height of the arc above the chord), and the area of the circular segment.
Distance from the center to the circle.
The angle at the center between the two radii to the chord's endpoints.
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Chord length
—
- Chord (exact)
- —
- Radius
- —
- Central angle
- —
- Sagitta (arc height)
- —
- Distance from center
- —
- Arc length
- —
More details (1 more)
- Segment area
- —
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Study path
Use this calculator with
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What the chord length calculator solves
The chord length calculator solves the chord of a circle three ways: from the radius and the central angle (c = 2r·sin(θ/2)), from the radius and the chord's perpendicular distance from the center (c = 2·sqrt(r² − d²)), or backward from a measured chord and sagitta to recover the radius. Every mode also reports the arc length, sagitta (height of the arc above the chord), and the area of the circular segment.
Formula
How to use it
- Pick the mode that matches what you measured: radius + angle, radius + distance from the center, or chord + sagitta.
- Enter the two values in one unit and select that unit.
- Read the chord length, then the sagitta, arc length, central angle, and segment area for the same arc.
How to read the answer
The chord is the straight line across the circle; the arc is the curved path between the same two points, so the arc is always longer. The sagitta is how far the arc bulges above the chord, and the segment area is the sliver between them. Distance from center plus sagitta always equals the radius.
Common mistakes and edge cases
- Using the full angle θ inside the sine — the formula needs half the angle: c = 2r·sin(θ/2).
- A distance from the center equal to or greater than the radius means the line misses the circle (or just touches it), so no chord exists.
- Mixing degrees and An angle unit where a full turn is 2π instead of 360 degrees.: enter the central angle in degrees; the calculator converts for the arc-length step.
- Confusing the sagitta (arc height) with the distance from the center; they add up to the radius.
Worked examples
Radius 10, central angle 60° (chord = radius)
Chord length
10
Radius 10, chord 6 from the center
Chord length
16
Chord 16 with sagitta 4 (radius 10)
Chord length
16
Radius 5, angle 180° (a diameter)
Chord length
10
Decimal radius 7.5, distance 0
Chord length
15
Distance equal to radius (impossible)
Chord length
Error
Angle 0° (no chord)
Chord length
Error
Frequently asked questions
What is the chord length formula?+
c = 2r·sin(θ/2) with the central angle θ, or c = 2·sqrt(r² − d²) with the chord's distance d from the center. Both come from the right triangle formed by the radius, half the chord, and d.
What is a sagitta?+
The height of an arc: the distance from the chord's midpoint up to the arc. It equals r − d, or r(1 − cos(θ/2)). Woodworkers and engineers measure it to find the radius of a curve.
How do I find the radius from a chord and its height?+
r = c²/(8h) + h/2. A chord of 16 with an arc height of 4 comes from a circle of radius 16²/32 + 2 = 10.
What is the longest chord of a circle?+
The diameter, 2r. It has central angle 180° and passes through the center (distance 0).
What is a circular segment?+
The region between a chord and its arc. Its area is (r²/2)(θ − sin θ) with θ in radians — the sector minus the triangle.
Is the chord longer or shorter than the arc?+
Shorter, always. A straight line is the shortest path between the endpoints; the arc curves away from it. They get closer as the angle shrinks.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026