Chord Of A Circle - Definition, Formula, Theorems, Example - Cuemath

Chord of Circle

The chord of a circle is defined as the line segment joining any two points on the circumference of the circle. It should be noted that the diameter is the longest chord of a circle that passes through the center of the circle.

1.	What is the Chord of a Circle?
2.	Properties of the Chord of a Circle
3.	Formula of Chord of Circle
4.	Theorems of Chord of a Circle
5.	FAQs on Chord of a Circle

What is the Chord of a Circle?

A line segment that joins two points on the circumference of the circle is defined as the chord of the circle. Among the other line segments that can be drawn in a circle, the chord is one whose endpoints lie on the circumference. Observe the following circle to identify the chord PQ. Diameter is also considered to be a chord which passes through the center of the circle.

Properties of the Chord of a Circle

Given below are a few important properties of the chords of a circle.

• The perpendicular to a chord, drawn from the center of the circle, bisects the chord.
• Chords of a circle, equidistant from the center of the circle are equal.
• There is one and only one circle which passes through three collinear points.
• When a chord of circle is drawn, it divides the circle into two regions, referred to as the segments of the circle: the major segment and the minor segment.
• A chord when extended infinitely on both sides becomes a secant.

Formula of Chord of Circle

There are two basic formulas to find the length of the chord of a circle:

• Chord length using perpendicular distance from the center = 2 × √(r2 − d2). Let us see the proof and derivation of this formula. In the circle given below, radius 'r' is the hypotenuse of the triangle that is formed. Perpendicular bisector 'd' is one of the legs of the right triangle. We know that the perpendicular bisector from the center of the circle to the chord bisects the chord. Therefore, half of the chord forms the other leg of the right triangle. By Pythagoras theorem, (1/2 chord)2 + d2 = r2, which further gives 1/2 of Chord length = √(r2 − d2). Thus, chord length = 2 × √(r2 − d2)

• Chord length using trigonometry = 2 × r × sin(θ/2); where 'r' is the radius of the circle and 'θ' is the angle subtended at the center by the chord. Observe the following circle to see the central angle 'θ' subtended by the chord AB and 'r' as the radius of the circle.

 

Theorems of Chord of a Circle

The chord of a circle has a few theorems related to it.

Theorem 1: The perpendicular to a chord, drawn from the center of the circle, bisects the chord.

Observe the following circle to understand the theorem in which OP is the perpendicular bisector of chord AB and the chord gets bisected into AP and PB. This means AP = PB

Theorem 2: Chords of a circle, equidistant from the center of the circle are equal.

Observe the following circle to understand the theorem in which chord AB = chord CD, and they are equidistant from the center if PO = OQ.

Theorem 3: For two unequal chords of a circle, the larger chord will be closer to the center than the smaller chord. (Unequal Chords Theorem)

If we draw multiple chords in a circle starting from the diameter to both the ends, we will observe that as we move closer to the center, the chord increases in length.

Important Notes

• The radius of a circle bisects the chord at 90°.
• When two radii join the two ends of a chord, they form an isosceles triangle.
• The diameter is the longest chord of a circle.

Related Links:

• Circles
• Area of a Circle
• Area of a Sector of a Circle
• Segment of a Circle
• Arc Length