Chord Length Calculator
A chord is a straight line connecting two points on a circle. This calculator computes the chord length and the segment height (the perpendicular distance from the chord to the arc) from the radius and the central angle between the two endpoints. The chord length formula uses trigonometry and depends on both the radius of the circle and the central angle. The segment height, also called the sagitta, measures how far the chord is from the center of the circle at its midpoint.
Chord formulas
Chord length = 2 * r * sin(theta / 2)
Segment height = r * (1 - cos(theta / 2))
where theta is the central angle in radians
Example calculation
For a chord with radius 10 and central angle of 60 degrees (1.047 radians):
- Chord length = 2 * 10 * sin(0.524) = 20 * 0.5 = 10.00 units
- Segment height = 10 * (1 - cos(0.524)) = 10 * 0.134 = 1.34 units
Chord length calculator: frequently asked questions
What is a chord?
A chord is a straight line connecting two points on a circle. The diameter is the longest chord. A chord is shorter than the arc that connects the same two points.
What is the formula for chord length?
The chord length is calculated as: Chord = 2 * r * sin(theta / 2), where r is the radius and theta is the central angle in radians.
What is the segment height (sagitta)?
The segment height, also called the sagitta, is the perpendicular distance from the midpoint of the chord to the arc. It is calculated as: Height = r * (1 - cos(theta / 2)).
How does chord length relate to the arc?
The chord is always shorter than the arc connecting the same two points. The chord length depends on the central angle and the radius. As the central angle increases, the chord length increases.
What are practical applications of chord calculations?
Chord calculations are used in engineering for designing arches, in architecture for curved structural elements, and in navigation for measuring distances along circular paths.
Official sources
- Wolfram MathWorld: Chord.
- ISO 80000-2: Mathematical signs and symbols.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.