Chord Length Calculator – Guide & Formulas
Calculate the length of a chord in a circle using the radius and distance from center, or radius and central angle. Step-by-step results.
The Chord Length Calculator computes the length of a chord in a circle using either the radius and perpendicular distance from center, or the radius and central angle. It also calculates related values like sagitta (arrow height) and arc length.
Key Takeaway
Use the free Chord Length Calculator to calculate the length of a chord in a circle using the radius and distance from center, or radius and central angle. step-by-step results. Get instant results with step-by-step explanations.
How to Use the Chord Length Calculator
- Choose your calculation mode: Radius + Distance, or Radius + Angle.
- Enter the radius of the circle.
- Enter the perpendicular distance from center (Mode 1) or the central angle in degrees (Mode 2).
- Review the chord length, sagitta, arc length, and segment area.
The Formula
Variable Definitions
- c: Length of the chord
- r: Radius of the circle
- d: Perpendicular distance from the center of the circle to the chord
- θ: Central angle subtended by the chord (in degrees or radians)
Example: Chord in a Circle with r=10, d=6
Find the chord length when the radius is 10 and the distance from center is 6.
- Step 1: Identify the formula: c = 2√(r² − d²).
- Step 2: Substitute values: c = 2√(10² − 6²).
- Step 3: Calculate inside the square root: 100 − 36 = 64.
- Step 4: Take the square root: √64 = 8.
- Step 5: Chord length: c = 2 × 8 = 16 units.
Frequently Asked Questions
What is a chord in a circle?
A chord is a straight line segment whose endpoints both lie on the circle. The diameter is the longest possible chord, passing through the center.
How do I calculate chord length from the radius and central angle?
Use c = 2r × sin(θ/2), where r is the radius and θ is the central angle in degrees. This formula derives from the isosceles triangle formed by the two radii and the chord.
What is the sagitta of a chord?
The sagitta (or arrow) is the perpendicular distance from the midpoint of the chord to the arc. It is calculated as s = r − √(r² − (c/2)²).
What is the maximum chord length?
The maximum chord length equals the diameter (2r), which occurs when the chord passes through the center (d = 0).
What happens when d = r?
When d = r, the chord length is 0. This means the chord is tangent to the circle—it touches the circle at exactly one point, so the two endpoints coincide.
How does chord length relate to arc length?
The chord is the straight-line distance between two points on the circle, while the arc is the curved distance along the circle. Arc length is always ≥ chord length.
What is the area of a circular segment?
A circular segment is the region between a chord and its arc. Its area equals the sector area minus the triangle area: A = ½r²(θ − sin θ), where θ is in radians.
Can I use this calculator for ellipses?
This calculator is designed for circles. Ellipse chord calculations require more complex formulas involving both semi-major and semi-minor axes.
What is the perpendicular bisector of a chord?
A line drawn from the center of a circle perpendicular to a chord always bisects the chord. This is a fundamental theorem of circle geometry.
How do I find the distance from center given chord length and radius?
Rearrange the formula: d = √(r² − (c/2)²). This gives the perpendicular distance from the center to the chord.
What is the relationship between two parallel chords?
Two parallel chords in a circle are equidistant from the center if and only if they have equal length. Unequal parallel chords are at different distances from center.
How is chord length used in music?
In acoustics, chord geometry relates to string vibration. The fundamental frequency of a vibrating string is inversely proportional to its length (chord length between fixed points).
What is the longest chord perpendicular to a diameter?
Any chord perpendicular to a diameter is bisected by that diameter. The longer the chord, the closer it is to the center. The diameter itself is the longest such chord.
How do I calculate chord length in construction?
Chord calculations are used in arch construction, curved walls, circular staircases, and pipe bending. Builders measure chord length to mark curved elements.
What is the chord length of a unit circle?
For a unit circle (r=1), the chord length is c = 2sin(θ/2), where θ is the central angle. For θ = 60°, c = 1 (equilateral triangle).
How does chord length change with angle?
As the central angle increases from 0° to 180°, the chord length increases from 0 to 2r (diameter). Beyond 180°, the chord length decreases symmetrically.
What is the relationship between chord and tangent?
A tangent line touches the circle at one point and is perpendicular to the radius at that point. The chord-tangent angle equals half the intercepted arc measure.
Can I calculate chord length for irregular curves?
For non-circular curves, the chord length is simply the straight-line distance between two points on the curve, calculated using the distance formula.
How is chord length used in surveying?
Surveyors use chord lengths to measure curved boundaries, road curves, and property lines along circular arcs. The chord-tangent method is standard in route surveying.
What is the versine of a chord?
The versine (versine = 1 − cos θ) relates to the sagitta: sagitta = r × versine(θ/2). It was historically used in astronomical calculations and navigation.