Chord of a Circle: Definition & Formula

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  • 0:00 What Is a Chord of a Circle?
  • 0:37 Examples
  • 1:23 Associated Vocabulary
  • 1:42 Formulas
  • 3:08 The Pythagorean Theorem
  • 4:26 Lesson Summary
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Lesson Transcript
Instructor
Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In this lesson, you'll learn the definition of a chord of a circle. You will also learn the formulas to find the chord of a circle and then look at some examples.

What is a Chord of a Circle?

Before we get into the actual definition of a chord of a circle, it may be helpful to visualize an example.

Imagine that you are on one side of a perfectly circular lake and looking across to a fishing pier on the other side. The chord is the line going across the circle from point A (you) to point B (the fishing pier). The circle outlining the lake's perimeter is called the circumference. A chord of a circle is a line that connects two points on a circle's circumference.

Examples

To illustrate further, let's look at several points of reference on the same circular lake from before. If each point of reference (i.e. duck feeding area, picnic tables, you, water fountain, and fishing pier) were directly on this lake's circumference, then each line connecting a point to another point on the circle would be chords.

In this image, we have added letters for each reference point, so we can easily label the chords.

  • The line between the fishing pier and you is now chord AC
  • The line between the water fountain and duck feeding area is now chord BE
  • The line between you and the picnic tables is chord CD

And so on…

Associated Vocabulary

If we had a chord that went directly through the center of a circle, it would be called a diameter. If we had a line that did not stop at the circle's circumference and instead extended into infinity, it would no longer be a chord; it would be called a secant.

Formulas

The formulas to find the length of a chord vary depending on what information about the circle you already know.

Formula 1: If you know the radius and the value of the angle subtended at the center by the chord, the formula would be:

We can use this diagram to find the chord length by plugging in the radius and angle subtended at the center by the chord into the formula. So, if we plug in the values of the radius and the angle measurement into a scientific calculator, we would get the chord length value as approximately 5.74.

Formula 2: If you know the radius and the perpendicular distance from the chord to the circle center, the formula would be:

Remember that d in this formula is the perpendicular distance from the chord to the center of the circle.

Here, we know the radius is 5 and the perpendicular distance from the chord to the center is 4. So, if we plug in the values of the radius and the perpendicular distance from the chord to the center of the circle, we would get the chord length value as 6.

The Pythagorean Theorem

If you look at formula 2, it is essentially a variation of the Pythagorean theorem. We can find the chord of a circle using formula 2, but we can also use the Pythagorean theorem. Let's look at this figure:

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Additional Activities

Using the Equations for the Length of a Chord to find other Values

In the video lesson we learned two equations that can be used to find the length, L, of a chord of a circle, L = 2rsin(theta/2), where r is the radius of the circle and theta is the angle subtended at the center by the chord, and L = 2 sqrt(r^2 - d^2), where r is the radius of the circle and d is the perpendicular distance between the chord and the center of the circle. We can use these same equation to find the radius of the circle, the perpendicular distance between the chord and the center of the circle, and the angle subtended at the center by the chord, provided we have enough information.

Practice Problems

1) If the length of a chord is 5 and the perpendicular distance between the chord and the center is 2, what is the radius of the circle?

2) If the length of a chord is 10 and the radius of the circle is 15, what is the angle subtended at the center by the chord?

3) If the angle subtended at the center by the chord is 60 degrees and the radius of the circle is 9, what is the perpendicular distance between the chord and the center of the circle?

Solutions

1) If the length of a chord is 5 and the perpendicular distance between the chord and the center is 2, what is the radius of the circle?

Since we know the length of the chord and the perpendicular distance between the chord and the center of the circle, we can find the radius of the circle using the equation L=2sqrt(r^2 - d^2) with L = 5 and d = 2.

L = 2sqrt(r^2 - d^2)

5 = 2sqrt(r^2 - 2^2)

2.5 = sqrt(r^2 - 4)

6.25 = r^2 - 4

10.25 = r^2

sqrt(10.25) = r

3.2 is approximately r

The radius is about 3.2.

2) If the length of a chord is 10 and the radius of the circle is 15, what is the angle subtended at the center by the chord?

Since we know the length of the chord and the radius and are trying to find the angle subtended at the center by the chord, we can use L = 2rsin(theta/2) with L = 10 and r = 15.

L = 2rsin(theta/2)

10 = 2(15)sin(theta/2)

10/30 = sin(theta/2)

1/3 = sin(theta/2)

sin^{-1}(1/3) = theta/2

2sin^{-1}(1/3) = theta

theta is approximately 38.94 degrees

The angle subtended at the center by the chord is about 38.94 degrees.

3) If the angle subtended at the center by the chord is 60 degrees, and the radius of the circle is 9, what is the perpendicular distance between the chord and the center of the circle?

We have to use both equations for this problem. First, we will use

L = 2rsin(theta/2)

to find the length of the chord, and then we can use L = 2sqrt(r^2 - d^2) to find the perpendicular distance between the chord and the center of the circle.

L = 2rsin(theta/2)

L = 2(9)sin(60/2)

L = 18sin(30)

L = 9

The length of the chord is 9 and so

L = 2sqrt(r^2 - d^2)

9 = 2sqrt(9^2 - d^2)

4.5 = sqrt(81 - d^2)

20.25 = 81 - d^2

d^2 = 60.75

d = sqrt(60.75)

d is approximately 7.79

The distance between the chord and the center of the circle is about 7.79.

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