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.

ChordDiagram

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.

CircularSectorCircularSegment

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.

ChordTheorems

In the left figure above,

 ab=cd
(1)

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

 PA·PB=PC·PD,
(2)

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).

Chord

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

 A=2int_0^(sqrt(R^2-r^2))x(y)dy.
(3)

But

 y^2+(r+x)^2=R^2,
(4)

so

 x(y)=sqrt(R^2-y^2)-r
(5)

and

A=2int_0^(sqrt(R^2-r^2))(sqrt(R^2-y^2)-r)dy
(6)
=R^2cos^(-1)(r/R)-rsqrt(R^2-r^2).
(7)

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

 A=1/2piR^2,
(8)

the expected area of the semicircle.

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