Chord Length Calculator

Calculate chord length, arc length, and segment area from the radius and central angle.

Enter Circle & Angle

Result

Chord Length
10
units
Chord Length (c)10
Arc Length (s)10.472
Segment Area9.0586
Sector Area52.3599
Sagitta (Height)1.3397
Angle in Radians1.0472

Step-by-Step Solution

c = 2r sin(θ/2)

Understanding Chord Length

A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The chord length depends on the radius of the circle and the central angle subtended by the chord. The diameter is the longest possible chord of any circle, corresponding to a central angle of 180 degrees.

The relationship between chord length, radius, and central angle is one of the fundamental results in circle geometry. It is derived from basic trigonometry by dropping a perpendicular from the center of the circle to the midpoint of the chord, creating two right triangles.

Key Formulas

Chord Length

The length of a chord given the radius r and central angle θ.

c = 2r · sin(θ/2)

Arc Length

The length of the arc between the two endpoints of the chord.

s = r · θ (radians)

Sector Area

The area of the pie-shaped sector formed by the central angle.

A_sector = (1/2) r^2 θ

Segment Area

The area between the chord and the arc (sector minus triangle).

A_seg = (r^2/2)(θ - sinθ)

Sagitta (Height)

The distance from the midpoint of the chord to the arc.

h = r(1 - cos(θ/2))

Apothem

The distance from the center of the circle to the midpoint of the chord.

d = r · cos(θ/2)

Derivation of the Chord Length Formula

Consider a circle with center O and radius r. Let A and B be two points on the circle such that the central angle AOB = θ. Drop a perpendicular from O to the chord AB, meeting it at point M. By symmetry, M is the midpoint of AB.

In the right triangle OMA, we have: sin(θ/2) = AM/r, therefore AM = r · sin(θ/2). Since M is the midpoint of AB, the full chord length is AB = 2 · AM = 2r · sin(θ/2).

Special Cases

  • θ = 60 degrees: The chord length equals the radius (c = r). The triangle formed is equilateral.
  • θ = 90 degrees: The chord length is r√2. This forms a right isosceles triangle with the center.
  • θ = 120 degrees: The chord length is r√3.
  • θ = 180 degrees: The chord is the diameter, c = 2r.

Applications of Chord Length

Chord length calculations are essential in many practical and theoretical contexts. Engineers use chord calculations when designing curved structures such as bridges, tunnels, and arches. In navigation, chord distance approximations are used for short-distance calculations on Earth's surface. Surveyors use chord-related formulas to lay out curved roadways and railway tracks.

  • Civil engineering: Designing curved roads, bridges, and tunnels requires precise chord and arc calculations.
  • Architecture: Arched windows, domes, and circular structures rely on chord geometry.
  • Navigation: Great circle distances on Earth can be approximated using chord length formulas.
  • Manufacturing: Cutting circular materials requires knowledge of chord lengths for precise measurements.
  • Computer graphics: Approximating circles with line segments (chords) is fundamental in rendering algorithms.

Tips for Accurate Calculations

  • Ensure you know whether your angle is in degrees or radians before calculating.
  • The central angle must be between 0 and 360 degrees (or 0 and 2π radians).
  • For angles greater than 180 degrees, the chord length decreases as the angle increases toward 360 degrees.
  • The maximum chord length (the diameter) occurs at exactly 180 degrees.