What Is the Chord Length Formula?

A chord of a circle is defined as a straight line segment whose endpoints both lie on the circle. The precise calculation of the length of a chord is a central technique in geometry and trigonometry, with its formulas having direct derivations from fundamental circle properties and coordinate geometry.

Geometric Definition of a Chord and its Length in a Circle

Let $O$ denote the center of a circle with radius $r$, and let $AB$ be any chord of this circle. The perpendicular distance from $O$ to $AB$ is denoted as $d$. The length of the chord $AB$ is denoted as $l$.

The line joining $O$ to the midpoint $M$ of $AB$ is perpendicular to $AB$. This construction divides the chord into two equal segments, each of length $\frac{l}{2}$, and creates a right triangle with legs of length $d$ and $\frac{l}{2}$, and hypotenuse of length $r$.

Explicit Derivation of the Chord Length—Distance from Center Given

By the Pythagorean theorem in triangle $OMA$: \[ OA^2 = OM^2 + AM^2 \] The lengths correspond to: \[ r^2 = d^2 + \left(\frac{l}{2}\right)^2 \] Isolating the term involving $l$: \[ \left(\frac{l}{2}\right)^2 = r^2 - d^2 \] Taking square roots on both sides gives: \[ \frac{l}{2} = \sqrt{r^2 - d^2} \] Multiplying both sides by $2$: \[ l = 2\sqrt{r^2 - d^2} \]

Result: The length of a chord at perpendicular distance $d$ from the center in a circle of radius $r$ is $l = 2\sqrt{r^2-d^2}$.

Chord Length Formula When Subtended Angle at Center is Known

Let the chord $AB$ subtend a central angle $\theta$ (in radians) at $O$. Draw radii $OA$ and $OB$; triangle $OAB$ is isosceles with $OA = OB = r$ and $\angle AOB = \theta$. The chord $AB$ forms the base of this triangle.

By applying the Law of Cosines in triangle $OAB$: \[ AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos\theta \] Substitute $OA = OB = r$: \[ AB^2 = r^2 + r^2 - 2r^2\cos\theta = 2r^2(1-\cos\theta) \] Take square roots: \[ AB = r\sqrt{2(1-\cos\theta)} \]

Alternatively, expressing in terms of the sine function using the identity $1-\cos\theta = 2\sin^2\left(\frac{\theta}{2}\right)$: \[ AB = r\sqrt{4\sin^2\left(\frac{\theta}{2}\right)} = 2r\sin\left(\frac{\theta}{2}\right) \]

Result: If a chord subtends angle $\theta$ at the center, then its length is $l=2r\sin\left(\frac{\theta}{2}\right)$.

Transformation Between Distance and Central Angle Formulations

Let the central angle subtended by chord $AB$ be $\theta$, and its perpendicular distance from the center be $d$. Draw the radius $OM$ perpendicular to $AB$ at $M$, the midpoint of $AB$. The triangle $OAM$ is right-angled at $M$, with $OA = r$, $OM = d$, and $AM = \frac{l}{2}$.

The half-angle $\frac{\theta}{2}$ at $O$ corresponds to triangle $OAM$, with: \[ \sin\left(\frac{\theta}{2}\right) = \frac{AM}{r} = \frac{l/2}{r} \] Which gives: \[ l = 2r\sin\left(\frac{\theta}{2}\right) \]

The value $d = r\cos\left(\frac{\theta}{2}\right)$, derived by \[ \cos\left(\frac{\theta}{2}\right) = \frac{OM}{OA} = \frac{d}{r} \] Therefore, \[ d = r\cos\left(\frac{\theta}{2}\right) \]

These relations allow conversion between distance-from-center and angle-based chord length forms.

Chord Length in Terms of Height (Sagitta) of the Chord

Let $h$ be the perpendicular distance from the chord to the arc (the height or sagitta of the chord). The construction forms a right-angled triangle joining the center $O$, the midpoint $M$ of $AB$, and an endpoint $A$, with $OM = r - h$, $OA = r$, $AM = \frac{l}{2}$.

By Pythagoras' theorem: \[ r^2 = (r-h)^2 + \left(\frac{l}{2}\right)^2 \] Expand $(r-h)^2$: \[ r^2 = r^2 - 2rh + h^2 + \left(\frac{l}{2}\right)^2 \] \[ 0 = -2rh + h^2 + \left(\frac{l}{2}\right)^2 \] \[ 2rh = h^2 + \left(\frac{l}{2}\right)^2 \] \[ 2rh - h^2 = \left(\frac{l}{2}\right)^2 \] \[ l = 2\sqrt{2rh - h^2} \]

Result: The chord length for known height $h$ is $l = 2\sqrt{2rh - h^2}$ where $r$ is the radius and $h$ is the sagitta.

Numerical Example—Chord Length by Distance from Center

Given: Radius $r=7\,\text{cm}$, Perpendicular distance from center to chord $d=4\,\text{cm}$.

Substitute in the formula $l=2\sqrt{r^2-d^2}$: \[ l = 2\sqrt{7^2-4^2} = 2\sqrt{49-16} = 2\sqrt{33} \] \[ \sqrt{33} \approx 5.74456 \] \[ l = 2 \times 5.74456 = 11.489\,\text{cm} \]

Final result: The chord length is $11.49\,\text{cm}$ (rounded to two decimal places).

Numerical Example—Chord Length by Central Angle (Degrees and Radians)

Given: Radius $r = 5$, Central angle $\theta = 30^\circ$.

First, convert $30^\circ$ to radians: \[ \theta = 30^\circ = \frac{\pi}{6}\,\text{radians} \] The chord length formula is $l = 2r\sin\left(\frac{\theta}{2}\right)$. Compute: \[ \frac{\theta}{2} = 15^\circ = \frac{\pi}{12}\,\text{radians} \] \[ l = 2 \times 5 \times \sin\left(15^\circ\right) \] \[ \sin(15^\circ) = \frac{\sqrt{6}-\sqrt{2}}{4} \] \[ l = 10 \times \frac{\sqrt{6}-\sqrt{2}}{4} \] \[ l = \frac{10}{4}(\sqrt{6}-\sqrt{2}) = 2.5(\sqrt{6}-\sqrt{2}) \] \[ \sqrt{6} \approx 2.4495,\quad \sqrt{2} \approx 1.4142 \] \[ 2.5 \times (2.4495-1.4142) = 2.5 \times 1.0353 = 2.58825 \]

Final result: The chord length is approximately $2.59$.

Reversing the Chord Length Formulas: Expressing Radius in Terms of Chord Length and Height

Given chord length $l$ and its height $h$, express the radius $r$ via the rearranged equation: \[ l = 2\sqrt{2rh-h^2} \] \[ \frac{l^2}{4} = 2rh-h^2 \] \[ 2rh = \frac{l^2}{4} + h^2 \] \[ r = \frac{l^2}{8h} + \frac{h}{2} \]

Result: The radius of the circle is $r = \frac{l^2}{8h} + \frac{h}{2}$ for a chord of length $l$ at sagitta $h$.

Special Case: The Diameter as the Longest Chord

For $\theta = \pi$ radians or $180^\circ$, the chord becomes the diameter. Then, \[ l = 2r\sin\left(\frac{\pi}{2}\right) = 2r \times 1 = 2r \] Thus, the diameter is always the longest possible chord of a circle, with length $2r$.

Related Concepts: Arc, Segment, and Sector—Precise Geometric Meaning

A segment of a circle is the region enclosed by a chord and the corresponding arc. The sector is the region bounded by two radii and the included arc. For derivations of area related to chords and segments, see Area Of Circle Formula.

Summary of Chord Length Formulas

Main formulas: $l = 2\sqrt{r^2-d^2}$, $l=2r\sin\left(\frac{\theta}{2}\right)$, $l = 2\sqrt{2rh-h^2}$, with reversal for $r$ as $r = \frac{l^2}{8h} + \frac{h}{2}$. These results form the foundation for problems in coordinate, trigonometric, and geometric contexts concerning chords of circles. For related information, refer to the page on Area And Perimeter Formula.