Chord Length Calculator
Find the length of a chord in a circle given the radius and the central angle it subtends. This calculator also computes the corresponding arc length for reference.
The Chord Length Formula
The chord length formula derives from basic trigonometry applied to the isosceles triangle formed by the two radii and the chord. When you draw radii to both endpoints of a chord, you create a triangle with two sides equal to r and the included angle equal to the central angle θ. Applying the law of cosines or using the sine function on the half-angle gives the formula: chord = 2r·sin(θ/2).
This formula works for any central angle from 0° to 360°. As the angle increases from 0°, the chord length grows, reaching its maximum of 2r (the diameter) at 180°. Beyond 180°, the formula still produces the correct chord length for the minor chord corresponding to the reflex angle. The simplicity of this formula makes it a staple of circle geometry calculations.
Chord vs Arc Length
While the chord is a straight line, the arc is the curved path along the circle between the same two points. Arc length is calculated as rθ where θ is in radians, or (θ/360°) × 2πr when using degrees. The arc is always longer than the chord for any angle less than 360°, reflecting the fundamental geometric principle that the shortest distance between two points is a straight line.
The relationship between chord and arc length matters in engineering and surveying. When building curved structures like bridges or tunnels, engineers need both the straight-line span (chord) and the curved distance (arc) to plan construction accurately. Surveyors use chord-to-arc conversions when measuring property boundaries that follow curved features like rivers or roads.
Practical Uses of Chord Calculations
Chord length calculations appear in many practical scenarios. In mechanical engineering, gear tooth profiles depend on chord measurements. In civil engineering, horizontal curves on roads and railways use chord lengths for layout and staking. In architecture, arched windows and doorways require precise chord measurements to ensure structural integrity and aesthetic proportions.
In navigation, the chord between two points on Earth's surface (modeled as a sphere) approximates the straight-line distance through the Earth, while the arc gives the surface distance. This distinction is important in aviation when calculating great circle routes versus direct tunnel distances. Even in music, the spacing of frets on stringed instruments involves chord-like calculations on circular geometry.
Frequently Asked Questions
What is a chord?
A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The diameter is the longest possible chord, passing through the center.
What is the chord length formula?
Chord length = 2r·sin(θ/2), where r is the radius and θ is the central angle in radians. For degrees, convert first by multiplying by π/180.
How is chord length different from arc length?
Chord length is the straight-line distance between two points on a circle. Arc length is the curved distance along the circumference between those same two points.
What angle gives the longest chord?
A central angle of 180° gives the longest chord, which is the diameter. The chord length equals 2r at this angle since sin(90°) = 1.
Can a chord be longer than the diameter?
No. The diameter is the maximum chord length. Any other chord will be shorter because it does not pass through the center of the circle.