Among properties of chords of a circle are the following:
The area that a circular chord "cuts off" is called a circular segment.
The chord function is defined geometrically as in the picture to the left. The chord of an angle is the length of the chord between two points on a unit circle separated by that angle. By taking one of the points to be zero, it can easily be related to the modern sine function:
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, not extant, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones:
|Pythagorean||sin2θ + cos2θ = 1|
The half-angle identity greatly expedites the creation of chord tables. Ancient chord tables typically used a large value for the radius of the circle, and reported the chords for this circle. It was then a simple matter of scaling to determine the necessary chord for any circle. According to G. J. Toomer, Hipparchus used a circle of radius 3438' (=3438/60=57.3). This value is extremely close to 180 / π (=57.29577951...). One advantage of this choice of radius was that he could very accurately approximate the chord of a small angle as the angle itself. In modern terms, it allowed a simple linear approximation: