Chord Length Formula

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.

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 the common chord of two circles formulas, chord radius formulas, etc.

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 Center 

Chord Length = \[2 \times \sqrt{r^2 - d^2} \]

Chord Length Formula Using Trigonometry

Chord Length =\[ 2 \times  r \times sin(\frac{c}{2}) \]

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

\[R = \frac{L^2}{8h} + \frac{h}{2} \]

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 circles formulas when the radius of two circles and distance between the center of the two circles is given below.

Length of the common chord of the two circle formula is:

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

Other Parts of a Circle

• The radius of Circle: The radius of a circle is described as the distance from the center to any other point on the boundary of the given circle. 

• Diameter of a Circle: The diameter of a circle is a straight line passing through the center of the circle that connects the 2 points on the boundary. The diameter of a given circle is always twice the radius of the given circle. 

• Circumference of Circle: The circumference of a circle is described as the perimeter of a circle. It is the distance around the boundary of the given circle. It is found by the formula- 

C = 2 × π × r (where r is the radius of the given circle)

• Area of a Circle - The area enclosed by a circle or the region that it occupies in a 2-Dimensional plane is called the area of the circle. Area of a given Circle) A = π × r2

• Arc of a Circle: An arc is the curve part or portion of a circle's circumference. Mathematically it can be calculated as - s = 2 π r (θ /3600) (where s is the length of the arc, r is radius of the circle and θ is the central angle of the circle)

• The Secant of Circles:  The secant of a circle is the line that crosses 2 points on the circumference of a circle. The word 'secant' literally means 'cut' in Latin. There exists a secant-tangent rule that states that,  when a secant line and a tangent of a given circle are constructed from a common exterior point, the multiplication of the secant and its external segment is always equal to the square of the tangent.

• Tangent: Tangent is the line that cuts the border of the circle exactly at 1 single point. The tangent point doesn't enter the interior of the circle. A circle can have infinite tangents as it is made up of infinite points. Tangents will always be perpendicular to the radius of the given circle.

• A Segment in a Circle: A segment of a circle is the area enclosed by a chord and the corresponding arc in a given circle. Segments are divided into 2 types -minor segment, and major segment. The Area of a segment = r2 (θ - sinθ) ÷ 2  (where r is the radius of the circle and θ is in radians)

•  A Sector of a Circle: The sector of a circle is the area enclosed by 2 radii and the corresponding arc. Sectors are divided into 2 types of sectors, minor sector, and major sector. If the sector cuts an angle θ (in radians) at the center, the area of a sector of that given circle = (θ x r2 ) ÷ 2 

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 \[\times\] 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, \[\angle POQ = 2 \times  \sqrt{PRQ}\]

∠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