<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

2.7: Length of a Chord

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated11 minsto complete
%
Progress
Practice Length of a Chord
Practice
Progress
Estimated11 minsto complete
%
Practice Now

You have been asked to help the younger students at your school with their Physical Education class. While working one afternoon, you are asked to take out a parachute that the students can play with. As the students are playing, one of them walks across a small portion of the parachute instead of under it like she is supposed to. If the chute is shaped like a circle with a radius of 6 meters, and the path the student walked across the chute covered an angle of \begin{align*}50^\circ\end{align*}50, what is the length of the path she walked across the parachute?

Read on, and at the completion of this Concept, you'll be able to answer this question.

Watch This

The first part of this video will help you understand what a chord is:

Chords

Guidance

You may recall from your Geometry studies that a chord is a segment that begins and ends on a circle.

\begin{align*}\overline{AB}\end{align*}AB¯¯¯¯¯¯¯¯ is a chord in the circle.

We can calculate the length of any chord if we know the angle measure and the length of the radius. Because each endpoint of the chord is on the circle, the distance from the center to \begin{align*}A\end{align*}A and \begin{align*}B\end{align*}B is the same as the radius length.

Next, if we bisect the angle, the angle bisector must be perpendicular to the chord and bisect it (we will leave the proof of this to your Geometry class). This forms a right triangle.

We can now use a simple sine ratio to find half the chord, called \begin{align*}c\end{align*}c here, and double the result to find the length of the chord.

\begin{align*}& \sin \frac{\theta}{2}=\frac{c}{r}\\ & c=r \times \sin \frac{\theta}{2}\end{align*}sinθ2=crc=r×sinθ2

So the length of the chord is:

\begin{align*}2c=2r \sin \frac{\theta}{2}\end{align*}2c=2rsinθ2

Example A

Find the length of the chord of a circle with radius 8 cm and a central angle of \begin{align*}110^\circ\end{align*}110. Approximate your answer to the nearest mm.

Solution: We must first convert the angle measure to radians:

\begin{align*}110 \times \frac{\pi}{180}=\frac{11\pi}{18}\end{align*}110×π180=11π18

Using the formula, half of the chord length should be the radius of the circle times the sine of half the angle.

\begin{align*}& \frac{11\pi}{18} \times \frac{1}{2}=\frac{11\pi}{36}\\ & 8 \times \sin \frac{11\pi}{36}\end{align*}11π18×12=11π368×sin11π36

Multiply this result by 2.

So, the length of the chord is approximately 13.1 cm.

Example B

Find the length of the chord of a circle with a radius of 2 m that has a central angle of \begin{align*}90^\circ\end{align*}90.

Solution: First convert the angle to radians:

\begin{align*}90 \times \frac{\pi}{180}=\frac{\pi}{2}\end{align*}90×π180=π2

Using the formula, half of the chord length should be the radius of the circle times the sine of half the angle.

\begin{align*}& \frac{\pi}{2} \times \frac{1}{2}=\frac{\pi}{4}\\ & 2 \times \sin \frac{\pi}{4}\end{align*}π2×12=π42×sinπ4

Multiply this result by 2.

So the answer is approximately 2.83 meters.

Example C

Find the length of the chord of a circle with radius 1 m and a central angle of \begin{align*}170^\circ\end{align*}170.

Solution: We must first convert the angle measure to radians:

\begin{align*}170 \times \frac{\pi}{180}=\frac{17\pi}{18}\end{align*}170×π180=17π18

Using the formula, half of the chord length should be the radius of the circle times the sine of half the angle.

\begin{align*}& \frac{17\pi}{18} \times \frac{1}{2}=\frac{17\pi}{36}\\ & 1 \times \sin \frac{17\pi}{36} = .996\end{align*}17π18×12=17π361×sin17π36=.996

Multiply this result by 2.

So, the length of the arc is approximately 1.992

Notice that the length of the chord is almost 2 meters, which would be the diameter of the circle. If the angle had been 180 degrees, the chord would have just been the distance all the way across the circle going through the middle, which is the diameter.

Vocabulary

Chord: A chord is a straight line across a circle, intersecting the circle in two places, but not passing through the circle's center.

Guided Practice

1. If you run a piece of string across a doughnut you are eating, and the radius between the endpoints of the string to the center of the doughnut is 4 inches, how long is the string if the angle swept out by the chord is \begin{align*}20^\circ\end{align*}20?

2. You are eating dinner one night with your family at the local Italian restaurant. A piece of spaghetti makes a chord across your plate. You know that the length of the spaghetti strand is 5 inches, and the radius of the plate is 7 inches. What is the angle swept out by the chord?

3. If you draw a chord across a circle and make a chord across it that has a length of 15 inches, sweeping out an angle of \begin{align*}\pi\end{align*}π radians, what is the radius of the circle you drew?

Solutions:

1. You can use the equation \begin{align*}C = 2r\sin \left( \frac{\theta}{2} \right)\end{align*}C=2rsin(θ2) to solve this problem: (Don't forget to convert angles to radians)

\begin{align*} C = 2r\sin \left( \frac{\theta}{2} \right)\\ C = (2)(4)\sin \left( \frac{.349}{2} \right)\\ C = 8(.1736)\\ C = 1.388\\ inches \end{align*}C=2rsin(θ2)C=(2)(4)sin(.3492)C=8(.1736)C=1.388inches

2. Since the radius of the plate and the length of the chord are known, you can solve for the angle:

\begin{align*} C = 2r\sin \left( \frac{\theta}{2} \right)\\ \frac{C}{2r} = \sin \left( \frac{\theta}{2} \right)\\ \sin^{-1} \left( \frac{c}{2r} \right) = \frac{\theta}{2}\\ \sin^{-1} \left( \frac{5}{14} \right) = \frac{\theta}{2}\\ .365 = \frac{\theta}{2}\\ \theta = .73\\ \end{align*}

The angle spanned by the spaghetti is .73 radians.

3. Using the equation for the length of a chord:

\begin{align*} c = 2r\sin \left( \frac{\theta}{2} \right)\\ 15 = (2r)\sin \left( \frac{\pi}{2} \right)\\ r = 7.5\\ \end{align*}

As you can see, the radius of the circle is 7.5 inches. This is what you should expect, since the chord sweeps out an angle of \begin{align*}\pi\end{align*}. This means that it sweeps out half of the circle, so that the chord is actually going across the whole diameter of the circle. So if the chord is going across the diameter and has a length of 15 inches, then the radius of the circle should be 7.5 inches.

Concept Problem Solution

With the equation for the length of a chord in hand, you can calculate the distance the student ran across the parachute:

First convert the measure in degrees to radians:

\begin{align*}50 \times \frac{\pi}{180} \approx .27\pi\end{align*}

\begin{align*}2r \sin \frac{\theta}{2} = (2)(6) \sin \frac{.27\pi}{2} = 12 \sin .135\pi \approx 4.94 meters\end{align*}

Practice

  1. Find the length of the chord of a circle with radius 1 m and a central angle of \begin{align*}100^\circ\end{align*}.
  2. Find the length of the chord of a circle with radius 8 km and a central angle of \begin{align*}130^\circ\end{align*}.
  3. Find the length of the chord of a circle with radius 4 in and a central angle of \begin{align*}45^\circ\end{align*}.
  4. Find the length of the chord of a circle with radius 3 ft and a central angle of \begin{align*}32^\circ\end{align*}.
  5. Find the length of the chord of a circle with radius 2 cm and a central angle of \begin{align*}112^\circ\end{align*}.
  6. Find the length of the chord of a circle with radius 7 in and a central angle of \begin{align*}135^\circ\end{align*}.

Solve for the missing variable in each circle.

Use the picture below for questions 13-15.

  1. Suppose you knew the length of the chord, the length of the radius, and the central angle of the above circle. Describe one way to find the length of the red segment using the Pythagorean Theorem.
  2. Suppose you knew the length of the chord, the length of the radius, and the central angle of the above circle. Describe one way to find the length of the red segment using cosine.
  3. What would you need to know in order to find the area of the segment (the portion of the circle between the chord and the edge of the circle)? Describe how to find the area of this region.

Vocabulary

Chord

Chord

A chord is a straight line across a circle, intersecting the circle in two places, but not passing through the circle's center.

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Subjects:
Grades:
Date Created:
Sep 26, 2012
Last Modified:
Mar 23, 2016
Files can only be attached to the latest version of Modality
Reviews
50 % of people thought this content was helpful.
1
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.TRG.227.L.1