# Chord (geometry)

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*.

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

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: