# Chord of a Circle

Sub Topics
A line segment whose end points lie on a circle is called a chord.

In the adjoining figure, each of the line segments PQ, RS, AB and CD is a chord of the circle with centre O. Clearly, an infinite number of chords may be drawn in a circle.

## What is a Chord of a Circle?

Two end points on the circumference of a circle joined together is known as chord of a circle. Here the diameter is the greatest chord of the circle.

## Chord of a Circle Formula

If ‘ r ’ is the radius of the circle and ‘ a ‘ is the length of the arc, then length of the chord made by the arc is 2r * Sin($\frac{\theta}{2}$)

Proof:

If ‘ r ’ is the radius of the circle and ‘ $\theta$ ‘ is the angle made by the chord at the centre of the circle and length of the arc ‘ a ‘
Arc AB subtends an angle ‘ $\theta$ ’ at the centre. Where ‘ $\theta$ ‘ is the measure of angle in radians.

The rule that involves length of the arc ‘ a ‘ and the central angle ‘ $\theta$ ’ and radius ‘ r ‘ is

a = r$\theta$ - - - - - - - - - - - - - - - - - - - (i)

AB is the chord of the circle, ‘ O ‘ is the centre of the circle, draw OM perpendicular to AB.

M is the midpoint of AB and OM bisects $\angle$ AOB.

Therefore, $\angle$ AOM = $\angle$ BOM = $\frac{\theta}{2}$

In triangle AMO, $\angle$ AMO = 900

Sin($\frac{\theta}{2}$) = $\frac{AM}{OA}$

Sin($\frac{\theta}{2}$) = $\frac{AM}{r}$

r * Sin($\frac{\theta}{2}$) = AM

Since the length of the chord AB = 2 * AM = 2r * Sin($\frac{\theta}{2}$)

Therefore, formula to find the length of the chord =   2r * Sin( $\frac{\theta}{2}$ )

## Chord Length of a Circle

Find the length of the chord of a circle if radius of the circle and central angle made by the chord are given.

Proof:

If ‘ r ’ is the radius of the circle and ‘ $\theta$ ‘ is the angle made by the chord at the centre of the circle.

AB is the chord of the circle, ‘ O ‘ is the centre of the circle, draw OM perpendicular to AB.

M is the midpoint of AB and OM bisects $\angle$ AOB.

Therefore, $\angle$ AOM = $\angle$ BOM = $\frac{\theta}{2}$

In triangle AMO, $\angle$ AMO = 900

Sin($\frac{\theta}{2}$) = $\frac{AM}{OA}$

Sin($\frac{\theta}{2}$) = $\frac{AM}{r}$

r * Sin($\frac{\theta}{2}$) = AM

Since the length of the chord AB = 2 * AM = 2r * Sin($\frac{\theta}{2}$)

Therefore, formula to find the length of the chord =   2r * Sin($\frac{\theta}{2}$)

## How to Find the Chord of a Circle

If ‘ r ‘ is the radius of the circle and ‘ p ‘ is the length of the perpendicular drawn to the chord from the centre of the circle then the length of the chord is given by 2

Proof:

Given, a circle of radius ‘ r ‘ and ‘ p ‘ be the length of the perpendicular drawn from centre ‘O’ on the chord.

AB is the chord of the circle, ‘ O ‘ is the centre of the circle, draw OM perpendicular to AB.

In triangle AMO, $\angle$ AMO = 900

Using Pythagoras theorem, we get

OA2 = AM2 + OM2

r2 = AM2 + p2

r2 - p2 = AM2

AM = 2

Since the length of the chord AB = 2 * AM = 2