Home / All Definitions / Geometry / Chord Definition

Chord Definition

A chord of a circle is a straight line segment on the interior of a circle whose endpoints both lie on that circle. The infinite line extension of a chord is a secant line, or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word chord is derived from the Latin term chorda meaning bowstring. The term is also used in graph theory, where a cycle chord of a graph cycle C is an edge not in C whose endpoints lie in C.


Among properties of chords of a circle are the following:

  • Chords are equidistant from the center if and only if their lengths are equal.

  • Equal chords are subtended by equal angles from the center of the circle.

  • A chord that passes through the center of a circle is called a diameter and is the longest chord.

  • If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).


The midpoints of a set of parallel chords of an ellipse are collinear.


Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7.5 degrees. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.

The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: crd θ =  (1 – cos θ)2 + sin2 θ  =  2 – 2cos θ  = 2 sin(θ2).

The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably a great deal was known about them. In the table below (where c is the chord length, and D the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones:





sin2 θ + cos2 θ = 1

crd2 θ + crd2 (π - θ) = 4


sin θ2 = ±  1 - cos θ2 

crd θ2 = ±  2 - crd(π - θ) 

Apothem (a)

c = 2 r2 - a2 

c =  D2 - 4a2 

Angle (θ)

c = 2r sin(θ2)

c = D2crd θ

Related Definitions


“Chord.” From Wolfram MathWorld, mathworld.wolfram.com/Chord.html.

“Chord (Geometry).” Wikipedia, Wikimedia Foundation, 7 May 2020, en.wikipedia.org/wiki/Chord_(geometry).


Welcome to Math Converse

Maintenance Icon

Our site is presently undergoing maintenance in order to upgrade the site.

Please be patient and understand that it will take some time to complete this work and random things may not work as intended.


App Icon

Check out our free app for iOS & Android.

For more information about our app visit here!

Browser Extension

Browser Extension Icon

Check out our free browser extension for Chrome, Firefox, Edge, Safari, & Opera.

For more information about our browser extension visit here!

Add to Home Screen

Add to Home Screen Icon

Add Math Converse as app to your home screen.

QR Code

Take a photo of the qr code to share this page or to open it quickly on your phone:

Cite This Page


Data Storage Showcase
Acceleration Showcase
Angle Showcase


Copy Link
Cite Page
Google Classroom
Google Bookmarks
Facebook Messenger
QR Code