Chord Length Calculator
Solve chord length problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Chord = 2r sin(ฮธ/2)
Where r is the radius of the circle and ฮธ is the central angle in radians subtended by the chord. Alternatively, given the sagitta s: Chord = 2โ(s(2r โ s)). The sagitta is calculated as s = r(1 โ cos(ฮธ/2)).
Worked Examples
Example 1: Chord from Central Angle
Problem:Find the chord length of a circle with radius 10 cm and a central angle of 60 degrees.
Solution:Convert angle to radians: 60 x pi/180 = 1.0472 rad\nChord = 2r sin(theta/2) = 2(10) sin(1.0472/2)\n= 20 x sin(0.5236)\n= 20 x 0.5\n= 10.0000 cm\nSagitta = r(1 - cos(theta/2)) = 10(1 - cos(0.5236)) = 10(1 - 0.8660) = 1.3397 cm
Result:Chord Length = 10.0000 cm | Sagitta = 1.3397 cm
Example 2: Chord from Sagitta
Problem:A circular arch has a radius of 20 m and a sagitta (rise) of 5 m. Find the chord length (span).
Solution:Chord = 2 x sqrt(s x (2r - s))\n= 2 x sqrt(5 x (40 - 5))\n= 2 x sqrt(5 x 35)\n= 2 x sqrt(175)\n= 2 x 13.2288\n= 26.4575 m\nCentral angle = 2 arcsin(chord / (2r)) = 2 arcsin(26.4575/40) = 2 arcsin(0.6614) = 82.82 degrees
Result:Chord Length = 26.4575 m | Central Angle = 82.82 degrees
Frequently Asked Questions
What is a chord of a circle and how is chord length calculated?
A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The chord length can be calculated using the formula: Chord Length = 2r sin(theta/2), where r is the radius and theta is the central angle in radians subtended by the chord at the center. The longest possible chord of any circle is its diameter, which passes through the center. Chords are fundamental in geometry, trigonometry, and engineering applications such as bridge arch design, gear tooth spacing, and circular tank construction. Understanding chord properties helps solve many practical measurement problems involving circular shapes.
What is the difference between a chord and an arc?
A chord is the straight-line distance between two points on a circle, while an arc is the curved path along the circumference between those same two points. For a given pair of points on a circle, the arc length is always greater than or equal to the chord length. They are equal only in the degenerate case when both points coincide. The relationship between chord length and arc length depends on the central angle: as the angle approaches zero, the chord and arc lengths become nearly equal. This distinction is critical in navigation, road design (curves), and manufacturing where both straight-line and curved measurements matter for accurate construction and planning.
What is the sagitta and how does it relate to chord length?
The sagitta (also called the versine or arc height) is the perpendicular distance from the midpoint of a chord to the arc of the circle. It can be calculated as s = r(1 - cos(theta/2)), where r is the radius and theta is the central angle. Given the sagitta and radius, you can find the chord length using: Chord = 2 * sqrt(s * (2r - s)). The sagitta is extremely useful in practical applications like determining the curvature of lenses in optics, calculating the rise of an arch in architecture, measuring the depth of a circular segment in tank volume calculations, and setting out curves in road and railway engineering.
How are chord lengths used in real-world engineering?
Chord lengths have numerous practical engineering applications. In civil engineering, chord measurements help lay out circular curves for highways and railways. In mechanical engineering, chord calculations are essential for gear tooth design, cam profiles, and pulley systems. Architects use chord geometry when designing arched structures, domes, and curved facades. In aerospace, chord length refers to the distance from the leading edge to the trailing edge of an airfoil or wing cross-section, which is critical for aerodynamic performance. Surveyors use chord measurements to establish circular boundaries and curved property lines. The concept also appears in music theory where chord spacing on circular instruments uses similar geometric principles.
References
Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy