A region of a circle contained by a line and a chord

In plane geometry, a chord is the line segment joining two points on a curve. The term is often used to describe a line segment whose ends lie on a circle.

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.


In the above figure, R is the radius of the circle, a is the chord length, r is called the apothem, and h the sagitta.


The shaded region in the left figure is called a circular sector, and the shaded region in the right figure is called a circular segment.

There are a number of interesting theorems about chords of circles. All angles inscribed in a circle and subtended by the same chord are equal. The converse is also true: The locus of all points from which a given segment subtends equal angles is a circle.


In the left figure above,


(Jurgensen 1963, p.345). In the right figure above,


which is a statement of the fact that the circle power is independent of the choice of the line ABP (Coxeter 1969, p.81; Jurgensen 1963, p.346).

Given any closed convex curve, it is possible to find a point P through which three chords, inclined to one another at angles of 60 degrees, pass such that P is the midpoint of all three (Wells 1991).


Let a circle of radius R have a chord at distance r. The area enclosed by the chord, shown as the shaded region in the above figure, is then








Checking the limits, when r=R, A=0 and when r-0,


the expected area of the semicircle.