Chord Calculator

Category: Geometry
- April 04, 2025
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Calculate various properties of a chord in a circle including chord length, arc length, chord height, and more. This calculator supports multiple calculation methods and provides visualisations with step-by-step explanations.

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About Chords in Geometry

A chord is a line segment whose endpoints both lie on a circle. Chords are important in geometry and have numerous applications in mathematics, physics, engineering, and more.

Key Properties and Formulas
  • Chord Length: c = 2r × sin(θ/2) [where r is radius and θ is central angle]
  • Chord Height: h = r - d = r(1 - cos(θ/2)) [where d is distance from chord to centre]
  • Arc Length: s = r × θ [where θ is in radians]
  • Distance from Centre: d = r × cos(θ/2)
  • Segment Area: A = (r²/2) × (θ - sin(θ)) [where θ is in radians]
Important Relationships
  • The perpendicular bisector of a chord passes through the centre of the circle.
  • Equal chords are equidistant from the centre.
  • Chords equidistant from the centre are equal in length.
  • If two chords intersect, the product of the segments of one chord equals the product of the segments of the other.
  • The chord length relates to the chord height by the formula: c² = 4r × h - 4h².
Applications

Chords have many practical applications:

  • Construction: Laying out circular arcs and curves
  • Engineering: Designing circular components and curves
  • Physics: Analysing circular motion and wave propagation
  • Music Theory: The term "chord" in music originates from geometry
  • Architecture: Planning arches and curved structures
  • Astronomy: Calculating distances and angles in celestial observations