# Arc Length

## Portion of a circle's circumference.

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Arc Length

A radian is another unit of measurement for angles just like degrees. Radians are especially convenient for measuring angles that have to do with circles. There are \begin{align*}360^\circ\end{align*} in a circle and \begin{align*}90^\circ\end{align*} in a right angle. How many radians are in a circle and how many radians make up a right angle?

### Arc Length

Recall that a portion of a circle is called an arc. One way to measure an arc is with degrees. The measure of an arc is equal to the measure of its corresponding central angle. Below, \begin{align*}m\widehat{DC}=70^\circ\end{align*} and \begin{align*}m\widehat{GH}=70^\circ\end{align*}.

When you measure an arc in degrees, it tells you the relative size of the arc compared to the whole circle. It does not tell you anything about the absolute size of the arc or the circle it came from. Both arcs above have the same measure, but \begin{align*}\widehat{GH}\end{align*} is physically longer, due to circle \begin{align*}E\end{align*} being bigger.

This leads to another way of describing the size of an arc. Arc length measures the distance (in units such as inches or centimeters) along a circle between the endpoints that define the arc. Above, \begin{align*}\widehat{GH}\end{align*} has a greater arc length than \begin{align*}\widehat{DC}\end{align*}. Because the radius of a circle is what determines the circle's size, the length of an arc intercepted by a given angle will be directly proportional to the radius of the circle. You will derive this fact in Examples A and B.

Arc length suggests a new way of measuring angles in circles. Previously, you have measured all angles in degrees, but you can also measure angles in radians. 1 radian is the angle that creates an arc that has a length equal to the radius. Below, the arc has a length equal to the radius. The angle that is created is 1 radian.

You can use the formula for the circumference of a circle to show that there are \begin{align*}2\pi\end{align*} radians in a circle (You will justify this in the Examples). This means that:

\begin{align*}2\pi \ radians=360^\circ\end{align*}

Therefore, you have the following conversions:

\begin{align*}1\ radian &=\left(\frac{180}{\pi}\right) degrees \ (\approx 57.3^\circ) \\ 1\ degree &=\left(\frac{\pi}{180}\right ) radians \ (\approx 0.17 \ radians) \end{align*}

Note that if a given angle has no units, it is assumed to be in radians.

A sector is a portion of a filled circle bounded by two radii and an arc (a “wedge” of a circle).

Let's do a problem involving sector similarity.

Show that two sectors with the same central angle are similar by using transformations to explain why the two shaded sectors below are similar.

Two shapes are similar if a similarity transformation exists between them. Draw a vector from point \begin{align*}A\end{align*} to point \begin{align*}E\end{align*}. Translate the red sector along the vector.

Rotate the image so that \begin{align*}\overline{EC^\prime}\end{align*} lies on \begin{align*}\overline{EH}\end{align*}. Because angles are preserved with translations and rotations, the image of \begin{align*}\overline{ED^\prime}\end{align*} will lie on \begin{align*}\overline{EG}\end{align*}.

Dilate the image about point \begin{align*}E\end{align*} by a factor of \begin{align*}\frac{EH}{AC}\end{align*}.

The two sectors are similar because a sequence of rigid transformations followed by a dilation carried one sector to the other.

Next, let's look at a problem involving arc length.

Explain why the length of arc \begin{align*}s\end{align*} is equal to \begin{align*}\theta r\end{align*}, where \begin{align*}\theta\end{align*} is the central angle in radians and \begin{align*}r\end{align*} is the radius of the circle.

In a circle of radius 1, the measure of a central angle in radians will be equal to the length of the intercepted arc. This is because the number of radians equals the number of radii that make up the arc.

If a sector has radius \begin{align*}r\end{align*}, it is similar to a sector of radius 1 with the same central angle (as shown in the previous problem about sectors).

Because the sectors are similar, corresponding lengths are proportional:

\begin{align*}\frac{r}{1} &=\frac{s}{x} \\ s &=xr\end{align*}

\begin{align*}x\end{align*} must be equal to the measure of \begin{align*}\theta\end{align*} in radians. Therefore:

\begin{align*}s=\theta r\end{align*}

Why does this make sense? Remember that if \begin{align*}\theta\end{align*} is in radians, then \begin{align*}\theta\end{align*} is equal to the number of radii that fit around the arc. The number of radii that fit around the arc multiplied by the length of the radius will equal the length of the arc.

Finally, lets look at a problem where we measure arc length

Find \begin{align*}m\widehat{GH}\end{align*} and the length of \begin{align*}\widehat{GH}\end{align*}.

\begin{align*}m\widehat{GH}=106^\circ\end{align*}. To find the length of the arc, multiply the radius (6 in) by the measure of the central angle in radians.

Remember that \begin{align*}1 \ degree=\left(\frac{\pi}{180}\right) \ radians\end{align*}. This means that \begin{align*}106^\circ=106\cdot \left(\frac{\pi}{180}\right)\approx 1.85 \ radians\end{align*}.

Now you can find the length of the arc:

\begin{align*}s &=\theta r\\ s &=(1.85)(6)\\ s & \approx 11.1\ in \end{align*}

Don't forget that your angle must be in radians in order to use the formula \begin{align*}s=\theta r\end{align*}!

### Examples

#### Example 1

There are \begin{align*}360^\circ\end{align*} in a circle and \begin{align*}90^\circ\end{align*} in a right angle. How many radians are in a circle and how many radians make up a right angle?

There are \begin{align*}2\pi\end{align*} radians in a circle. A right angle is \begin{align*}90^\circ\end{align*} and \begin{align*}90^\circ\end{align*} is \begin{align*}\frac{1}{4}\end{align*} of a circle, so there are \begin{align*}\frac{2\pi}{4}=\frac{\pi}{2}\end{align*}  radians in a right angle.

#### Example 2

Explain why a circle has \begin{align*}2 \pi \ radians\end{align*}.

The circumference of a circle with radius 1 is \begin{align*}2 \pi(1)=2 \pi\end{align*}. Therefore, \begin{align*}2 \pi\end{align*} radii fit around a circle with radius 1. All circles are similar, so \begin{align*}2 \pi\end{align*} radii must fit around all circles. 1 radian is the angle that creates an arc that has a length equal to the radius, so \begin{align*}2 \pi\end{align*} radians is the angle that creates an arc with a length equal to \begin{align*}2 \pi\end{align*} radii. Therefore, a circle is \begin{align*}2 \pi\end{align*} radians because \begin{align*}2 \pi\end{align*} radii fit around any circle.

#### Example 3

How many radians are there in \begin{align*}150^\circ\end{align*}?

Remember that \begin{align*}1 \ degree = \left(\frac{\pi}{180}\right) \ radians\end{align*}. This means that \begin{align*}150^\circ = 150 \cdot \left(\frac{\pi}{180}\right) \approx 2.62 \ radians\end{align*}.

#### Example 4

Find the length of \begin{align*}\widehat{CD}\end{align*}.

\begin{align*}s = \theta r\end{align*} and \begin{align*}\theta = 150^\circ=2.62 \ radians\end{align*}.  The length of \begin{align*}\widehat{CD}\end{align*} is \begin{align*}s = (2.62)(4) \approx 10.48 \ cm\end{align*}.

### Review

1. What's the difference between finding the measure of an arc and the length of an arc?

2. What is the relationship between a radian and a radius?

3. How are radians and degrees related?

4. Explain why the two shaded sectors below are similar.

5. Justify why the length of arc \begin{align*}s\end{align*} is equal to \begin{align*}\theta r\end{align*}, where \begin{align*}\theta\end{align*} is the central angle in radians and \begin{align*}r\end{align*} is the radius of the circle.

Convert each angle measured in degrees to an angle measured in radians. Leave answers in terms of \begin{align*}\pi\end{align*}.

6. \begin{align*}180^\circ\end{align*}

7. \begin{align*}360^\circ\end{align*}

8. \begin{align*}90^\circ\end{align*}

9. \begin{align*}60^\circ\end{align*}

10. \begin{align*}30^\circ\end{align*}

Find the measure in degrees and length in centimeters of \begin{align*}\widehat{CD}\end{align*} in each circle.

11.

12.

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15. Explain how to find the length of an arc when given the central angle in radians. How does this compare to the process of finding the length of an arc when given the central angle in degrees?

To see the Review answers, open this PDF file and look for section 8.9.

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### Vocabulary Language: English

TermDefinition
Arc An arc is a section of the circumference of a circle.
arc length In calculus, arc length is the length of a plane function curve over an interval.
Circumference The circumference of a circle is the measure of the distance around the outside edge of a circle.
Dilation To reduce or enlarge a figure according to a scale factor is a dilation.
radian A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.
Sector A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees.