Chord Length Calculator
Solve chord length problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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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)).
Last reviewed: December 2025
Worked Examples
Example 1: Chord from Central Angle
Example 2: Chord from Sagitta
Background & Theory
The Chord Length Calculator โ Circle Geometry applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Chord Length Calculator โ Circle Geometry traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
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)).
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.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
References
Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy