Chord Calculator

Enter a circle's radius and central angle to get the chord length, arc length, sagitta (height), and segment area using the standard circle-chord formulas.

Quick Facts

Formula
Chord length c = 2r × sin(θ/2)
r is the radius; θ is the central angle in radians.
Special case
At θ = 180°, the chord equals the diameter (2r)
This is the longest possible chord in the circle.

Your Results

Calculated
Chord length
-
Straight-line distance between the chord's endpoints
Arc length
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Distance along the curved arc
Sagitta (height)
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Chord midpoint to arc midpoint
Segment area
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Area enclosed between chord and arc

Ready

Enter a radius and central angle, then press Calculate.

Understanding the Chord Calculator

A chord is a straight line segment that connects two points on a circle's circumference. This calculator uses a circle's radius (r) and the central angle (θ) that the chord subtends at the center to compute the chord's length, along with three related circle-segment measurements: arc length, sagitta, and segment area.

The formulas

All four results follow directly from the radius and the central angle expressed in radians:

  • Chord length: c = 2r × sin(θ/2) — the straight-line distance between the two points where the chord meets the circle.
  • Arc length: s = r × θ — the distance along the curved path between the same two points.
  • Sagitta (height): h = r × (1 − cos(θ/2)) — the distance from the midpoint of the chord to the midpoint of the arc.
  • Segment area: A = (r² / 2) × (θ − sin θ) — the area enclosed between the chord and the arc.

If your angle is in degrees, convert it first: θ(radians) = θ(degrees) × π / 180. Selecting "Degrees" as the angle unit does this conversion automatically before the formulas run.

Common reference points

  • θ = 60°: the chord length equals the radius exactly (c = r), since sin(30°) = 0.5.
  • θ = 90°: the chord length is r√2 ≈ 1.414r.
  • θ = 180°: the chord passes through the center and equals the diameter (c = 2r) — the longest possible chord.
  • θ close to 0°: chord length and arc length converge, since a very small arc is nearly straight.

Reading the segment type

A chord divides a circle into two regions called segments. When the central angle is under 180°, the smaller region is the minor segment and the larger region on the other side is the major segment. At exactly 180°, the chord is a diameter and splits the circle into two equal semicircles. The calculator labels which case applies to your inputs after you press Calculate.

Frequently Asked Questions

What is a chord in geometry?
A chord is a straight line segment whose two endpoints both lie on a circle's circumference. The diameter is the longest possible chord — the special case where the chord passes through the center, corresponding to a central angle of 180°.
How do I find the length of a chord?
Multiply twice the radius by the sine of half the central angle: c = 2r × sin(θ/2), with θ in radians. If your angle is in degrees, convert it to radians first (multiply by π/180), or simply select "Degrees" in this calculator and it converts automatically.
What is the sagitta of a chord?
The sagitta is the perpendicular distance from the midpoint of the chord to the midpoint of the arc it cuts off — essentially how far the arc bulges beyond the straight chord. It equals h = r × (1 − cos(θ/2)) and is used in fields like optics and arch or bridge design.
What is the difference between a chord and an arc?
A chord is the straight-line segment connecting two points on a circle, while an arc is the curved path along the circle between those same two points. The arc is always longer than its chord, except in the limiting case of a vanishingly small angle where the two lengths converge.

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