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Chord Length Calculator

A chord connects two points on a circle's circumference without necessarily passing through the center. Engineers, architects, and geometry students need to find chord lengths when designing arcs, building curved structures, or solving circle problems. This calculator determines chord length from multiple parameters—central angle, chord height, or apothem—and also computes related circle properties like arc length, sector area, and segment area.

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Creators Mateusz Mucha, BEng
4,768 people find this calculator helpful

Understanding Chords in Circles

A chord is a straight line segment joining any two distinct points on a circle's perimeter. Unlike a diameter, which always passes through the center, a chord can connect any two circumference points at the same radial distance from the circle's center.

Key relationships:

  • The radius extends from the center to any circumference point
  • The apothem is the perpendicular distance from the center to the chord's midpoint
  • The chord height (sagitta) measures the maximum perpendicular distance from the chord to the arc
  • The central angle is the angle subtended by the chord at the circle's center

These relationships create multiple paths to calculate chord length, depending on which parameters you know.

Chord Length Formulas

Three primary equations govern chord length calculations. Select the one matching your available data:

Chord length = 2 × radius × sin(angle ÷ 2)

Chord length = 2 × √(radius² − apothem²)

Chord length = 2 × √(2 × radius × chord height − chord height²)

  • radius — Distance from the circle's center to any point on the circumference
  • angle — Central angle in radians or degrees subtended by the chord
  • apothem — Perpendicular distance from center to the chord's midpoint
  • chord height — Maximum perpendicular distance from the chord to the arc (sagitta)

Once you have the radius, you can compute additional circle properties:

  • Diameter: diameter = 2 × radius
  • Circumference: circumference = 2π × radius
  • Area: area = π × radius²
  • Arc length: arc length = radius × angle (in radians)
  • Sector area: sector area = (radius² × angle) ÷ 2
  • Segment area: segment area = (radius² ÷ 2) × (angle − sin(angle))

The segment area—the region between the chord and the arc—is particularly useful in construction and manufacturing when material volume or weight estimates are needed.

Common Mistakes and Practical Considerations

Avoid these pitfalls when calculating chord lengths:

  1. Angle unit mismatch — Many formulas require angles in radians, not degrees. If using the sine formula with degrees, ensure your calculator is in degree mode. Converting: radians = degrees × (π ÷ 180).
  2. Confusing chord height with radius — Chord height (sagitta) is not the same as radius. It's the perpendicular distance from the chord to the arc's highest point. Mixing these values will produce incorrect results.
  3. Apothem and distance terminology — The apothem equals radius minus chord height. If you know one, you can calculate the other. Verify which parameter your data actually represents before plugging numbers into formulas.
  4. Rounding intermediate results — In multi-step calculations, preserve decimal precision throughout. Only round the final answer. Rounding intermediate values compounds errors, especially in square root and trigonometric operations.

Chord vs. Arc: A Practical Distinction

A chord is the straight-line distance between two circumference points, while an arc is the curved path along the circle itself. For the same two endpoints, the arc is always longer than the chord—a fundamental property in circular geometry.

In practical applications, this matters:

  • Structural design: Use chord length for material spans; arc length for curved surface coverings
  • Measurement: Straight measuring tapes give chord length; calipers tracking the curve give arc length
  • Clearance calculations: Chord length determines straight-line obstruction; arc length determines path distance

A worked example: a circle with radius 5 and central angle 30° has a chord length of approximately 2.59 and an arc length of about 2.62 (in the same units).

Frequently Asked Questions

How do I calculate chord length when I only know the radius and central angle?

Apply the formula: chord length = 2 × radius × sin(angle ÷ 2). For example, with radius 5 and a 30° angle, convert 30° to radians (π ÷ 6 ≈ 0.524) or use degree mode. Then: chord length = 2 × 5 × sin(15°) ≈ 2.59 units. This formula works for any angle up to 180° and is the quickest method when the central angle is known.

What if I know the radius and chord height but not the angle?

Use the chord height formula: chord length = 2 × √(2 × radius × chord height − chord height²). For instance, with radius 10 and chord height 3, you get chord length = 2 × √(60 − 9) = 2 × √51 ≈ 14.3 units. This approach is common in manufacturing and construction when the arc's visual rise is measurable but the central angle is not readily available.

Why would I use the apothem instead of other parameters?

The apothem—perpendicular distance from center to chord midpoint—is often the easiest measurement in practical settings. Use: chord length = 2 × √(radius² − apothem²). If you know radius 8 and apothem 6, then chord length = 2 × √(64 − 36) = 2 × √28 ≈ 10.58 units. Surveyors and engineers frequently work with perpendicular offsets, making this formula valuable in the field.

How does the arc length differ from the chord length for the same two points?

Arc length travels along the curved circumference, while chord length spans the straight line between endpoints. Arc length = radius × angle (in radians), whereas chord length = 2 × radius × sin(angle ÷ 2). The arc is always longer. For a radius of 5 and 30° angle, arc length ≈ 2.62 and chord length ≈ 2.59. The difference grows larger with bigger central angles, reaching equality only as the angle approaches 0°.

Can this calculator determine the radius if I know the chord length?

Yes. Rearranging the formulas allows you to solve for radius given chord length and one other parameter (angle, apothem, or chord height). If you know chord length 10 and central angle 60°, then radius = chord length ÷ (2 × sin(angle ÷ 2)) = 10 ÷ (2 × sin(30°)) = 10 units. The calculator's 'Show other parameters' option displays additional input fields to make this reverse calculation seamless.

What is the segment area, and when would I need it?

The segment area is the region between the chord and the arc—the curved 'cap' of the circle. Formula: segment area = (radius² ÷ 2) × (angle − sin(angle)). This is essential for calculating material quantity, fluid volume in partially filled pipes, or weight distribution in curved structural members. A larger central angle increases segment area significantly, so the calculation is critical for engineering accuracy.

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