6.0: Arclength and Radians

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Imagine that you are riding on a Ferris wheel of radius 100 feet, and each rotation takes eight minutes. We can use angles in a standard position to describe your location as you travel around the wheel. The figure at right shows the locations indicated by $\theta = 0^{\circ}, 90^{\circ}, 180^{\circ}$, and $270^{\circ}$. But degrees are not the only way to specify a location on a circle.

$Screen Shot 2023-01-05 at 1.38.53 AM.png$

We could use percent of one complete rotation and label the same locations by $p = 0, p = 25, p = 50$, and $p = 75$. Or we could use the time elapsed, so that for this example we would have $t = 0, t = 2, t = 4$, and $t = 6$ minutes.

Another useful method uses distance traveled, or arclength, along the circle. How far have you traveled around the Ferris wheel at each of the locations shown?

Arclength

Recall that the circumference of a circle is proportional to its radius,

$C=2 \pi r$

If we walk around the entire circumference of a circle, the distance we travel is $2 \pi$ times the length of the radius, or about $6.28$ times the radius. If we walk only part of the way around the circle, then the distance we travel depends also on the angle of displacement.

For example, an angle of $45^{\circ}$ is $\frac{1}{8}$ of a complete revolution, so the arclength, $s$, from point $A$ to point $B$ in the figure below is $\frac{1}{8}$ of the circumference. Thus

$Screen Shot 2023-01-05 at 1.44.00 AM.png$

$s=\dfrac{1}{8}(2 \pi r)=\dfrac{\pi}{4} r$

Similarly, the angle of displacement from point $A$ to point $C$ is $\frac{3}{4}$ of a complete revolution, so the arclength along the circle from $A$ to $C$, shown below, is

$Screen Shot 2023-01-05 at 1.45.40 AM.png$

$s=\dfrac{3}{4}(2 \pi r)=\dfrac{3\pi}{2} r$

In general, for a given circle the length of the arc spanned by an angle is proportional to the size of the angle.

Arclength on a Circle.

$\text { Arclength }=\text { (fraction of one revolution) } \cdot(2 \pi r)$

The Ferris wheel in the introduction has circumference

$C=2 \pi(100)=628 \text { feet }$

so in half a revolution you travel 314 feet around the edge, and in one-quarter revolution you travel 157 feet.

To indicate the same four locations on the wheel by distance traveled, we would use $s=0, s=157, s=314, \text { and } s=471,$ as shown below.

$Screen Shot 2023-01-05 at 1.49.40 AM.png$

Example 6.1

What length of arc is spanned by an angle of $120^{\circ}$ on a circle of radius 12 centimeters?

Answer

Because $\frac{120}{360}=\frac{1}{3}$, an angle of $120^{\circ}$ is $\frac{1}{3}$ of a complete revolution, as shown below.

$Screen Shot 2023-01-05 at 1.55.45 AM.png$

Using the formula above with $r=12$, we find that

$s=\dfrac{1}{3}(2 \pi \cdot 12)=\dfrac{2 \pi}{3} \cdot 12=8 \pi \mathrm{cm}$

or about 25.1 cm.

Checkpoint 6.2

How far have you traveled around the edge of a Ferris wheel of radius 100 feet when you have turned through an angle of $150^{\circ}$?

Answer: 261.8 ft

Measuring Angles in Radians

If you think about measuring arclength, you will see that the degree measure of the spanning angle is not as important as the fraction of one revolution it covers. This observation suggests a new unit of measurement for angles, one that is better suited to calculations involving arclength. We’ll make one change in our formula for arclength, from

Arclength = (fraction of one revolution) · $(2\pi r)$

Arclength = (fraction of one revolution $\times 2\pi$) · $r$

We’ll call the quantity in parentheses, (fraction of one revolution $\times 2\pi$), the radian measure of the angle that spans the arc.

Radians.

The radian measure of an angle is given by

(fraction of one revolution $\times 2\pi$)

For example, one complete revolution, or $360^{\circ}$, is equal to $2 \pi$ radians, and one-quarter revolution, or $90^{\circ}$, is equal to $\frac{1}{4}(2 \pi)$ or $\frac{\pi}{2}$ radians. The figure below shows the radian measure of the quadrantal angles.

$Screen Shot 2023-01-05 at 2.07.41 AM.png$

Example 6.3

What is the radian measure of an angle of $120^{\circ}$?

Answer: An angle of $120^{\circ}$ is $\frac{1}{3}$ of a complete revolution, as we saw in the previous example. Thus, an angle of $120^{\circ}$ has a radian measure of $\frac{1}{3}(2 \pi)$, or $\frac{2 \pi}{3}$.

Checkpoint 6.4

What fraction of a revolution is $\pi$ radians? How many degrees is that?

Answer: Half a revolution, $180^{\circ}$

Radian measure does not have to be expressed in multiples of $\pi$. Remember that $\pi \approx 3.14$, so one complete revolution is about 6.28 radians, and one-quarter revolution is $\frac{1}{4}(2 \pi)=\frac{\pi}{2}$, or about 1.57 radians. The figure below shows decimal approximations for the quadrantal angles.

$Screen Shot 2023-01-05 at 2.14.55 AM.png$

Degrees	Radians: Exact Values	Radians: Decimal Approximations
$0^{\circ}$	0	0
$90^{\circ}$	$\dfrac{\pi}{2}$	1.57
$180^{\circ}$	$\pi$	3.14
$270^{\circ}$	$\dfrac{3\pi}{2}$	4.71
$360^{\circ}$	$2\pi$	6.28

Note 6.5 You should memorize both the exact values of these angles in radians and their approximations!

Example 6.6

In which quadrant would you find an angle of 2 radians? An angle of 5 radians?

Answer: Look at the figure above. The second quadrant includes angles between $\frac{\pi}{2}$ and $\pi$, or 1.57 and 3.14 radians, so 2 radians lies in the second quadrant. An angle of 5 radians is between 4.71 and 6.28, or between $\frac{3 \pi}{2}$ and $2 \pi$ radians, so it lies in the fourth quadrant.

Checkpoint 6.7

Draw a circle centered at the origin and sketch (in standard position) angles of approximately 3 radians, 4 radians, and 6 radians.

Answer: $Screen Shot 2023-01-05 at 2.20.04 AM.png$

Converting Between Degrees and Radians

It is not difficult to convert the measure of an angle in degrees to its measure in radians, or vice versa. One complete revolution is equal to 2 radians or to $360^{\circ}$, so

$2 \pi \text { radians }=360^{\circ}$

Dividing both sides of this equation by 2 gives us a conversion factor :

Unit Conversion for Angles.

$\dfrac{180^{\circ}}{\pi \text { radians }}=\mathbf{1}$

Note 6.8

To convert from radians to degrees we multiply the radian measure by $\frac{180^{\circ}}{\pi}$.
To convert from degrees to radians we multiply the degree measure by $\frac{\pi}{180}$.

Example 6.9

a Convert 3 radians to degrees.

b Convert 3 degrees to radians.

Answer: a $(3 \text{ radians }) \times\left(\dfrac{180^{\circ}}{\pi}\right)=\dfrac{540^{\circ}}{\pi} \approx 171.9^{\circ}$
b $\left(3^{\circ}\right) \times\left(\dfrac{\pi}{180^{\circ}}\right)=\dfrac{\pi}{60} \approx 0.05 \text{ radians}.$

Checkpoint 6.10

a Convert $60^{\circ}$ to radians. Give both an exact answer and an approximation to three decimal places.

b Convert $\frac{3 \pi}{4}$ radians to degrees.

Answer: a $\dfrac{\pi}{3} \approx 1.047$
b $135^{\circ}$

From our conversion factor we also learn that

$1 \text { radian }=\dfrac{180^{\circ}}{\pi} \approx 57.3^{\circ}$

$Screen Shot 2023-01-05 at 2.33.44 AM.png$

So while $1^{\circ}$ is a relatively small angle, 1 radian is much larger nearly $60^{\circ}$, in fact.

But this is reasonable, because there are only a little more than 6 radians in an entire revolution. An angle of 1 radian is shown above.

We’ll soon see that, for many applications, it is easier to work entirely in radians. For reference, the figure below shows a radian protractor.

$Screen Shot 2023-01-05 at 2.34.56 AM.png$

Arclength Formula

Measuring angles in radians has the following advantage: To calculate an arclength we need only multiply the radius of the circle by the radian measure of the spanning angle, $\theta$. Look again at our formula for arclength:

Arclength = ( fraction of one revolution $\times 2 \pi) \cdot r$

The quantity in parentheses, fraction of one revolution $\times 2 \pi$, is just the measure of the spanning angle in radians. Thus, if $\theta$ is measured in radians, we have the following formula for arclength, $s$.

Arclength Formula.

On a circle of radius $r$, the length $s$ of an arc spanned by an angle $\theta$ in radians is

$s=r \theta$

In particular, if $\theta=1$ we have $s=r$. We see that an angle of one radian spans an arc whose length is the radius of the circle. This is true for a circle of any size, as illustrated at right: an arclength equal to one radius determines a central angle of one radian, or about $57.3^{\circ}$.

$Screen Shot 2023-01-05 at 2.51.51 AM.png$

In the next example, we use the arclength formula to compute a change in latitude on the Earth’s surface. Latitude is measured in degrees north or south of the equator.

Example 6.11

The radius of the Earth is about 3960 miles. If you travel 500 miles due north, how many degrees of latitude will you traverse?

Answer

We think of the distance 500 miles as an arclength on the surface of the Earth, as shown at right. Substituting $s=500$ and $r=3960$ into the arclength formula gives

\begin{aligned}
500 & =3960 \theta \\
\theta & =\dfrac{500}{3960}=0.1263 \text { radians }
\end{aligned}

$Screen Shot 2023-01-05 at 2.54.09 AM.png$

To convert the angle measure to degrees, we multiply by $\dfrac{180^{\circ}}{\pi}$ to get

$0.1263\left(\dfrac{180^{\circ}}{\pi}\right)=7.23^{\circ}$

Your latitude has changed by about $7.23^{\circ}$.

Checkpoint 6.11

The radius of the Earth is about 3960 miles. If you travel 500 miles due north, how many degrees of latitude will you traverse?

Answer: 9.55 ft

Unit Circle

On a unit circle, $r=1$, so the arclength formula becomes $s=\theta$. Thus, on a unit circle, the measure of a (positive) angle in radians is equal to the length of the arc it spans.

Example 6.13

You have walked 4 miles around a circular pond of radius one mile. What is your position relative to your starting point?

Answer: The pond is a unit circle, so you have traversed an angle in radians equal to the arc length traveled, 4 miles. An angle of 4 radians is in the middle of the third quadrant relative to your starting point, more than halfway but less than three-quarters around the pond.

Checkpoint 6.14

An ant walks around the rim of a circular birdbath of radius 1 foot. How far has the ant walked when it has turned through an angle of $210^{\circ}$?

Answer: 3.67 ft

Review the following skills you will need for this section.

Algebra Refresher 6.1

Use the appropriate conversion factor to convert units.

1 $\dfrac{1 \text{ mile}}{1.609 \text{ kilometers}} = 1$

a 10 miles = ______ km

b 50 km = ______ miles

2 \(\dfrac{1 \text{ acre}}{0.405 \text{ hectare}} = 1

a 40 acres = ______ hectares

b 5 hectares = ______ acres

3 $\dfrac{1 \text{ horsepower}}{746 \text{ watts}} = 1$

a 250 horsepower = ______ watts

b 1000 watts = ______ horsepower

4 \(\dfrac{ 1 \text{ troy ounce}}{480 \text{ grains}} = 1

a 0.5 troy oz = ______ grains

b 100 grains = ______ troy oz

Algebra Refresher Answers

1 a. 16.09 km b. 31.08 mi

2 a. 16.2 hectares b. 12.35 acres

3 a. 186,500 watts b. 1.34 horsepower

4 a. 240 grains b. 0.21 troy oz

Section 6.1 Summary

Vocabulary

• Arclength

• Radian

• Conversion factor

• Latitude

• Unit circle