Chord Length Formula

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What is Chord in Circle?

The chord of a circle can be stated as a line segment joining two points on the circumference of the circle. The diameter is the longest chord of the circle which passes through the center of the circle. The figure shown below represents the circle and its chord.

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In the circle above with center O, AB represents the diameter of the circle (longest chord of a circle), OE represents the radius of a circle and CD represents the chord of a circle.

Let us consider CD as the chord of a circle and points P and Q lying anywhere on the circumference of the circle. If the two endpoints of the chord CD meet at point P, then ∠CPD is known as the angle extends by the chord CD at point P. The angle ∠CQD  is known as the angle extended by the chord CD at point Q.  The angle ∠CPD is known as the angle  extended by the chord CD at point P.

In this article, we will study what is a  chord in a circle, chord length formulas, how to find the length of the chord, length of common chord of two circles formulas, chord radius formulas, etc.


Angle Subtended by Chord

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How to Find the Length of the Chord?

There are two important formulas to find the length of the chords. The formula for the length of a chord is given as:

Chord Length Formula Using Perpendicular Distance from the Centre 

Chord Length = 2 × √(r² - d²)

Chord Length Formula Using Trigonometry

Chord Length = 2 × r × sin(c/2)

In the above formula for the length of a chord,

R represents the radius of the circle

C represents the angle extended at the center by the chord.

D represents the perpendicular distance from the cord to the center of the circle.


Chord Radius Formula

The chord radius formula when length and height of the chord are given is

R= L² / 8h + h/2

In the above chord radius formula,

R is the radius of a circle

L is the length of the chord

h is the height of th chord


Length of Common Chord of Two Circles Formula

The length of the common chord of two the circles formulas when radius of two circles and distance between the center of the two circles is given below.

Length of common chord of two circle formula is:

2 × radius 1 × radius 2 ÷ Distance between the center of two circles


Solved Examples

1. Calculate the length of the chord where the radius of the circle is 7cm and the perpendicular distance drawn from the center of the circle to its chord is 4 cm.

Solution:

Given,

Radius of a circle - 7 cm

And Distance, d  = 4 cm

Length of Chord Formula Circle = 2\[\sqrt{r^2-d^2}\]

Chord length = 2\[\sqrt{7^2-4^2}\]

Chord length = 2\[\sqrt{49 -16}\]

Chord length = 2\[\sqrt{33}\]

Chord length = 2 × 5.744

Or chord length  = 11.488 cm

2. In the circle given below, find the measure of ∠POQ when the value of ∠PRQ is given as 62°.

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Solution:

According to the property of chords of a circle, the angle extended at the center of the circle and an arc is twice the angle extended by it at any point on the circumference. 

Hence, ∠POQ is equal to twice of ∠PRQ. Hence, ∠POQ = 2 x \[\sqrt{PRQ}\]

∠POQ = 2 x 62° = 124°.


Quiz Time 

1. Find the length of the chord in the above- given circle

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  1. 5

  2. 4

  3. 7

  4. 6

2.  The longest chord of the circle is

  1. Radius

  2. Diameter

  3. Segment

  4. Arc

3.  If chords PQ and RS of congruent circles subtend equal angles at their centers, then:

  1.  PQ = RS

  2. PQ > RS

  3. PQ < RS

  4. None of the above

FAQ (Frequently Asked Questions)

1. What are the properties of the chords of a Circle?

Ans.

  • The chords which are equal in size cross equal angles at the center.

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  • The two chords which cross equal angles at the  center are equal.

  • A perpendicular drawn from the  center of the circle divides the chords. It implies both halves of the chord are similar in length.

  • Only one circle can travel through three noncollinear points.

  • Equal chords of a circle are at equal distance from the  center of the circle. 

  • The two chords that cross over equal distance from the center of the circle are equal in length.

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  • Angles drawn from the same center are always equal in proportions.

  • If the line segment connecting any two points crossing over identical angles at the two other points that are on the same side, they are considered as  concyclic. It implies that all fall on a similar circle.

  • The angle crossed over by an arc at the centre of the circle is twice the angle crossed over at any other given point on the circle.

  • The line which is formed from the center of a circle and that is bisecting the chord is perpendicular to the chord.

2. What are the different methods for finding the Length of the Chord?

Ans. There are two methods to find the length of the chord  depending on the information given in the questions. The two methods are:

  • When the radius and a central angle of a circle are given in the question, the length of the chord can be  calculated using the below formula:

Chord Length - 2 r sin(c/2)

Where r is the radius of a circle and c is the angle subtended at the center.

  • When the radius and distance of the center of a circle are given, the following formula can be applied.

Chord Length - 2√(r2 -d2)

Where r is the radius of the circle and d is the perpendicular distance of the center of the circle of the chord.