Chords of a Circle – Explanation & Examples

Chords of a CircleIn this article, you’ll learn:

  • What a chord of a circle is.
  • Properties of a chord and; and
  • How to find the length of a chord using different formulas.

 

What is the Chord of a Circle?

By definition, a chord is a straight line joining 2 points on the circumference of a circle. The diameter of a circle is considered to be the longest chord because it joins to points on the circumference of a circle.

In the circle below, AB, CD, and EF are the chords of the circle. Chord CD is the diameter of the circle.

What is the Chord of a Circle

Properties of a Chord

  • The radius of a circle is the perpendicular bisector of a chord.

The radius of a circle is the perpendicular bisector of a chord.

  • The length of a chord increases as the perpendicular distance from the center of the circle to the chord decreases and vice versa.
  • The diameter is the longest chord of a circle, whereby the perpendicular distance from the center of the circle to the chord is zero.
  • Two radii joining the ends of a chord to the center of a circle form an isosceles triangle.

Two radii joining the ends of a chord to the center of a circle forms an isosceles triangle.

  • Two chords are equal in length if they are equidistant from the center of a circle. For example, chord AB is equal to chord CD if PQ = QR.

Two chords are equal in length if they are equidistant from the center of a circle

How to Find the Chord of a Circle?

There are two formulas to find the length of a chord. Each formula is used depending on the information provided.

  • The length of a chord, given the radius and distance to the center of a circle.

If the length of the radius and distance between the center and chord is known, then the formula to find the length of the chord is given by,

Length of chord = 2√ (r2 – d2)

Where r = the radius of a circle and d = the perpendicular distance from the center of a circle to the chord.

The length of a chord given the radius and distance to center of a circle.

In the above illustration, the length of chord PQ = 2√ (r2 – d2)

  • The length of a chord, given the radius and central angle

If the radius and central angle of a chord are known, then the length of a chord is given by,

Length of a chord = 2 × r × sine (C/2)

= 2r sine (C/2)

Where r = the radius of the circle

C = the angle subtended at the center by the chord

d = the perpendicular distance from the center of a circle to the chord.

The length of a chord given the radius and central angle

Let’s work out a few examples involving the chord of a circle.

Example 1

The radius of a circle is 14 cm, and the perpendicular distance from the chord to the center is 8 cm. Find the length of the chord.

Solution

Given radius, r = 14 cm and perpendicular distance, d = 8 cm,

By the formula, Length of chord = 2√(r2−d2)

Substitute.

Length of chord = 2√ (142−82)

= 2√ (196 − 64)

= 2√ (132)

= 2 x 11.5

= 23

So, the length of the chord is 23 cm.

Example 2

The perpendicular distance from the center of a circle to the chord is 8 m. Calculate the chord’s length if the circle’s diameter is 34 m.

Solution

Given the distance, d = 8 m.

Diameter, D = 34 m. So, radius, r = D/2 = 34/2 = 17 m

Length of chord = 2√(r2−d2)

By substitution,

Length of chord = 2√ (172 − 82)

= 2√ (289 – 64)

= 2√ (225)

= 2 x 15

= 30

So, the length of the chord is 30 m.

Example 3

The length of a chord of a circle is 40 inches. Suppose the perpendicular distance from the center to the chord is 15 inches. What is the radius of the chord?

Solution

Given, length of chord = 40 inches.

Distance, d = 15 inches

Radius, r =?

By the formula, Length of chord = 2√(r2−d2)

40 = 2√ (r2 − 152)

40 = 2√ (r2 − 225)

Square both sides

1600 = 4 (r2 – 225)

1600 = 4r2 – 900

Add 900 on both sides.

2500 = 4r2

Dividing both sides by 4, we get,

r2 = 625

√r2 = √625

r = -25 or 25

Length can never be a negative number, so we pick positive 25 only.

Therefore, the radius of the circle is 25 inches.

Example 4

Given that radius of the circle shown below is 10 yards and the length of PQ is 16 yards. Calculate the distance OM.

Lenght from the center to the chord

Solution

PQ = length of chord = 16 yards.

Radius, r = 10 yards.

OM = distance, d =?

Length of chord = 2√(r2−d2)

16 =2√ (10 2− d 2)

16 =2√ (100 − d 2)

Square both sides.

256 = 4(100 − d 2)

256 = 400 − 4d2

Subtract 400 on both sides.

-144 = − 4d2

Divide both sides by -4.

36 = d2

d = -6 or 6.

Thus, the perpendicular distance is 6 yards.

Example 5:

Calculate the length of the chord PQ in the circle shown below.

Length of chord

Solution

Given the central angle, C = 800

The radius of the circle, r = 28 cm

Length of chord PQ =?

By the formula, length of chord = 2r sine (C/2)

Substitute.

Length of chord = 2r sine (C/2)

= 2 x 28 x Sine (80/2)

= 56 x sine 40

= 56 x 0.6428

= 36

Therefore, the length of the chord PQ is 36 cm.

Example 6

Calculate the length of the chord and the central angle of the chord in the circle shown below.

length of chord and the central angle

Solution

Given,

Perpendicular distance, d = 40 mm.

Radius, r = 90 mm.

Length of chord = 2√(r2−d2)

= 2√ (902 − 402)

= 2 √ (8100 − 1600)

= 2√6500

= 2 x 80.6

= 161.2

So, the length of the chord is 161.2 mm

Now calculate the angle subtended by the chord.

Length of chord = 2r sine (C/2)

161.2 = 2 x 90 sine (C/2)

161.2 = 180 sine (C/2)

Divide both sides by 180.

0.8956 = sine (C/2)

Find the sine inverse of 0.8956.

C/2 = 63.6 degrees

Multiply both sides by 2

C = 127.2 degrees.

So, the central angle subtended by the chord is 127.2 degrees.

 

Practice Questions

1. The radius of a circle is $26$ inches, and the perpendicular distance from the chord to the center is $10$ inches. What is the length of the chord?

2. The perpendicular distance from the center of a circle to the chord is $9$ feet. What is the chord’s length if the circle’s diameter is $82$ feet?

3. The length of a chord of a circle is $70$ inches. Suppose the perpendicular distance from the center to the chord is $12$ inches. What is the radius of the chord?

4. Given that radius of the circle shown below is $39$ mm long and the length of $AB$ $72$ mm. What is the length of $OB$?

Chords of a Circle 1

5. What is the length of the chord $MN$ in the circle shown below?

finding the length of a chord


 

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