A **chord** of a circle is a straight line segment whose endpoints both lie on a circular arc. 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 from the Latin *chorda* meaning *bowstring*.

## In circles

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).

## In ellipses

The midpoints of a set of parallel chords of an ellipse are collinear.^{[1]}

## In trigonometry

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+1/2 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 to 180 degrees by increments of 1/2 degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.^{[2]}

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:^{[2]}

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:

Name | Sine-based | Chord-based |
---|---|---|

Pythagorean | ||

Half-angle | ||

Apothem (
a) |
||

Angle (
θ) |

The inverse function exists as well:^{[3]}

## See also

- Circular segment - the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
- Scale of chords
- Ptolemy's table of chords
- Holditch's theorem, for a chord rotating in a convex closed curve
- Circle graph
- Exsecant and excosecant
- Versine and haversine
- Zindler curve (closed and simple curve in which all chords that divide the arc length into halves have the same length)

## References

**^**Chakerian, G. D. (1979). "7". In Honsberger, R. (ed.).*A Distorted View of Geometry*.*Mathematical Plums*. Washington, DC, USA: Mathematical Association of America. p. 147.- ^
^{a}^{b}Maor, Eli (1998),*Trigonometric Delights*, Princeton University Press, pp. 25–27, ISBN 978-0-691-15820-4 **^**Simpson, David G. (2001-11-08). "AUXTRIG" (FORTRAN-90 source code). Greenbelt, Maryland, USA: NASA Goddard Space Flight Center. Retrieved 2015-10-26.

## Further reading

- Hawking, Stephen William, ed. (2002).
*On the Shoulders of Giants: The Great Works of Physics and Astronomy*. Philadelphia, USA: Running Press. ISBN 0-7624-1698-X. LCCN 2002100441. Retrieved 2017-07-31. - Stávek, Jiří (2017-03-10) [2017-02-26]. "On the Hidden Beauty of Trigonometric Functions".
*Applied Physics Research*. Prague, CZ: Canadian Center of Science and Education.**9**(2): 57–64. doi:10.5539/apr.v9n2p57. ISSN 1916-9639. ISSN 1916-9647. [1]

## External links

Wikimedia Commons has media related to .Chord (geometry) |

- History of Trigonometry Outline
- Trigonometric functions, focusing on history
- Chord (of a circle) With interactive animation