Chord Length Calculator
Calculate chord length from radius and central angle or sagitta. Find arc length, sector area, segment area, and apothem with instant results.
Chord Length Calculator
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Formula: Chord = 2r sin(θ/2)
Worked example — Chord Length = 10.0000 cm | Sagitta = 1.3397 cm
Formula
Chord = 2r sin(θ/2)
Where r is the radius of the circle and θ is the central angle in radians subtended by the chord. Alternatively, given the sagitta s: Chord = 2√(s(2r − s)). The sagitta is calculated as s = r(1 − cos(θ/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 Chord = 2r sin(theta/2) = 2(10) sin(1.0472/2) = 20 x sin(0.5236) = 20 x 0.5 = 10.0000 cm Sagitta = 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)) = 2 x sqrt(5 x (40 - 5)) = 2 x sqrt(5 x 35) = 2 x sqrt(175) = 2 x 13.2288 = 26.4575 m Central 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.
Can you calculate chord length if you only know the arc length?
Yes, but it requires an iterative or numerical approach since there is no simple closed-form solution. If you know the arc length (L) and radius (r), you can first find the central angle: theta = L/r. Then apply the chord formula: Chord = 2r sin(theta/2). For example, with an arc length of 15.708 and radius 10, theta = 15.708/10 = 1.5708 radians (90 degrees), and the chord = 2(10)sin(0.7854) = 14.142. If you only know the arc length and chord length but not the radius, the problem becomes more complex and typically requires solving a transcendental equation numerically. Many engineering calculators and software packages include built-in functions for these calculations.
What is the apothem of a chord and how does it relate to the radius?
The apothem here is the perpendicular distance from the centre of the circle to the chord, so it is the radius minus the sagitta: a = r - s, or equivalently a = r cos(theta/2). Because that perpendicular bisects the chord, the apothem, the half-chord, and the radius form a right triangle, giving (Chord/2)^2 + a^2 = r^2. With r = 10 and a central angle of 90 degrees the half-chord is 7.0711 and the apothem is also 7.0711, and 7.0711^2 + 7.0711^2 = 100 as expected.
What is the difference between the sector area and the segment area?
The sector is the pie-slice bounded by two radii and the arc, with area ½r²theta for theta in radians. The segment is the smaller region bounded by the chord and the arc, which is the sector with the central triangle removed. Since that triangle has area ½ x Chord x apothem = ½r² sin(theta), the segment area is ½r²(theta - sin theta). Use the sector when you need the full wedge, such as a pie chart slice, and the segment when you need the liquid in a partly filled horizontal cylinder.
How do I find the radius of an arc from a chord and sagitta measured on site?
Rearranging Chord = 2 sqrt(s(2r - s)) for r gives r = Chord² / (8s) + s / 2, which needs only a taut string across the arc and one perpendicular measurement to the curve. For an arch spanning 26.4575 m with a rise of 5 m, r = 700 / 40 + 2.5 = 17.5 + 2.5 = 20 m. Surveyors, stonemasons, and sheet-metal fabricators rely on this because the centre of the circle is usually inaccessible or does not physically exist on the workpiece.
Why do two chords the same distance from the centre always have the same length?
Each chord forms a right triangle with the radius and its own apothem, so the half-chord is always sqrt(r² - a²). That value depends only on the perpendicular distance a, not on the chord's direction, so equal distances give equal lengths and the converse holds too. It also explains the monotonic pattern: as a chord moves away from the centre its apothem grows and its length shrinks, reaching zero as it becomes tangent, while a chord through the centre has apothem zero and becomes the diameter, the longest chord possible.
What happens to the chord when the central angle is greater than 180 degrees?
The chord starts getting shorter again, because 2r sin(theta/2) peaks at theta = 180 degrees where sin is 1 and the chord equals the diameter. A 270 degree angle gives the same chord as a 90 degree angle, since sin(135 degrees) and sin(45 degrees) are both 0.7071. What changes is which pieces you are describing: with a reflex angle the arc and sector are the major ones, the sagitta exceeds the radius, and the region cut off by the chord is the major segment. Remember to convert degrees to radians, theta_rad = theta_deg x pi / 180, before using any arc or area formula.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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