TOPICS
Search

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.

CircularSector
CircularSegment

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.


See also

Annulus, Apothem, Arc, Bertrand's Problem, Chordal Theorem, Circle Power, Circular Sector, Circular Segment, Concentric Circles, Cycle Chord, Holditch's Theorem, Radius, Sagitta, Secant Line, Semicircle

Explore with Wolfram|Alpha

References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Jurgensen, R. C.; Donnelly, A. J.; and Dolciani, M. P. Th. 42 in Modern Geometry: Structure and Method. Boston, MA: Houghton-Mifflin, 1963.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 29, 1991.

Referenced on Wolfram|Alpha

Chord

Cite this as:

Weisstein, Eric W. "Chord." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Chord.html

Subject classifications