Arclength on a Circle.
[latex]\text{Arclength}~ = \bigg( \text{ fraction of one revolution} \bigg) \cdot \bigg (2\pi r \bigg)[/latex]
Chapter 6: Radians
6.1 Arclength and Radians
Algebra Refresher
Algebra Refresher
Use the appropriate conversion factor to convert units.
1. [latex]\dfrac{1~ {mile}}{1.609~{kilometers}} = 1[/latex]
- 10 miles = km
- 50 km = miles
2. [latex]\dfrac{1~ {acre}}{0.405~{hectare}} = 1[/latex]
- 40 acres = hectares
- 5 hectares = acres
3. [latex]\dfrac{1~ {horsepower}}{746~{watts}} = 1[/latex]
- 250 horsepower = watts
- 1000 watts = horsepower
4. [latex]\dfrac{1~ {troy ounce}}{480~{grains}} = 1[/latex]
- 0.5 troy oz = grains
- 100 grains = troy oz
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Algebra Refresher Answers:
- a. [latex]16.09[/latex] km b. [latex]31.08[/latex] mi
- a. [latex]16.2[/latex] hectares b. [latex]12.35[/latex] acres
- a. [latex]186,500[/latex] watts b. [latex]1.34[/latex] horsepower
- a. [latex]240[/latex] grains b. [latex]0.21[/latex] troy oz
Learning Objectives
- Express angles in degrees and radians.
- Sketch angles given in radians.
- Estimate angles in radians.
- Use the arclength formula.
- Find coordinates of a point on a unit circle.
- Calculate angular velocity and area of a sector.
Arclength and Radians
Imagine that you are riding on a Ferris wheel of radius 100 feet, and each rotation takes eight minutes. We can use angles in standard position to describe your location as you travel around the wheel. The figure at right shows the locations indicated by [latex]\theta = 0°,~ 90°,~ 180°,[/latex] and [latex]270°{.}[/latex] But degrees are not the only way to specify location on a circle.
We could use percent of one complete rotation and label the same locations by [latex]p = 0,~ p = 25,~ p = 50,~{and}~ p = 75{.}[/latex] Or we could use the time elapsed, so that for this example, we would have [latex]t = 0,~ t = 2,~t = 4,~{and}~ t = 6[/latex] minutes. Another useful method to describe your location uses the distance traveled, or arclength, along the circle. How far have you traveled around the Ferris wheel at each of the locations shown? Before we consider that question, let’s agree on some vocabulary. An arc is a portion of a circle, and its length, quite naturally, is called arclength. An angle with vertex at the center of the circle is called a central angle, and a central angle whose sides meet the endpoints of an arc is said to subtend the arc. Or we may say that the angle spans the arc. If the arc represents a distance traveled, we sometimes refer to such an angle as the angle of displacement.
Arclength
Recall that the circumference of a circle is proportional to its radius,
[latex]C = 2 \pi r[/latex]
If we walk around the entire circumference of a circle, the distance we travel is [latex]2\pi[/latex] 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 [latex]45°[/latex] is [latex]\dfrac{1}{8}[/latex] of a complete revolution, so the the length of the arc from point [latex]A[/latex] to point [latex]B{,}[/latex] called [latex]s[/latex] in the figure at right, is [latex]\dfrac{1}{8}[/latex] of the circumference. Thus
[latex]s = \dfrac{1}{8}(2\pi r) = \dfrac{\pi}{4} r[/latex]
Similarly, the angle of displacement from point [latex]A[/latex] to point [latex]C[/latex] is [latex]\dfrac{3}{4}[/latex] of a complete revolution, so the arclength [latex]s[/latex] along the circle from [latex]A[/latex] to [latex]C{,}[/latex] shown at right, is
[latex]s = \dfrac{3}{4}(2\pi r) = \dfrac{3\pi}{2} r[/latex]
In general, for a given circle the length of the arc spanned by an angle is proportional to the size of the angle.
The Ferris wheel in the introduction has circumference
[latex]C \approx 2\pi (100) = 628~ {feet}[/latex]
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
[latex]s = 0,~ s \approx 157,~ s \approx 314,~ {and}~ s \approx 471{,}[/latex]
as shown at right.
Example 6.1.
What length of arc is spanned by an angle of [latex]120°[/latex] on a circle of radius 12 centimeters?
Solution
Because [latex]\dfrac{120}{360} = \dfrac{1}{3}{,}[/latex] an angle of [latex]120°[/latex] is [latex]\dfrac{1}{3}[/latex] of a complete revolution, as shown at right.
Using the formula above with [latex]r = 12{,}[/latex] we find that
[latex]s = \dfrac{1}{3}(2\pi \cdot 12) = \dfrac{2 \pi}{3} \cdot 12 = 8\pi ~ {cm}[/latex]
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 [latex]150°{?}[/latex]
Solution
Because the Ferris wheel has circumference
[latex]C = 2\pi (100) \approx 628~ {feet}[/latex],
using the formula above with [latex]r = 100{,}[/latex] we find that
[latex]s \approx \dfrac{5}{6} \cdot 314 = 261.67 ft[/latex]
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
[latex]\text{Arclength}~ = \bigg( \text{ fraction of one revolution} \bigg) \cdot \bigg (2\pi r \bigg)[/latex]
to
[latex]\text{Arclength}~ = \bigg( \text{ fraction of one revolution} \times 2\pi \bigg) \cdot r[/latex]
We’ll call the quantity in parentheses (fraction of one revolution [latex]\times 2\pi[/latex]) the radian measure of the angle that spans the arc.
Radians.
The radian measure of an angle is given by
[latex]\bigg( \text{fraction of one revolution}\times 2\pi \bigg)[/latex]
For example, one complete revolution, or [latex]360°{,}[/latex] is equal to [latex]2\pi[/latex] radians, and one-quarter revolution, or [latex]90°{,}[/latex] is equal to [latex]\dfrac{1}{4}(2\pi)[/latex] or [latex]\dfrac{\pi}{2}[/latex] radians. The figure below shows the radian measure of the quadrantal angles.
Example 6.3.
What is the radian measure of an angle of [latex]120°{?}[/latex]
Solution
An angle of [latex]120°[/latex] is [latex]\dfrac{1}{3}[/latex] of a complete revolution, as we saw in the previous example. Thus, an angle of [latex]120°[/latex] has a radian measure of [latex]\dfrac{1}{3}(2\pi){,}[/latex] or [latex]\dfrac{2\pi}{3}{.}[/latex]
Checkpoint 6.4.
What fraction of a revolution is [latex]\pi[/latex] radians? How many degrees is that?
Solution
Half a revolution, [latex]180°[/latex]
Radian measure does not have to be expressed in multiples of [latex]\pi{.}[/latex] Remember that [latex]\pi \approx 3.14{,}[/latex] so one complete revolution is about 6.28 radians, and one-quarter revolution is [latex]\dfrac{1}{4}(2\pi) = \dfrac{\pi}{2}{,}[/latex] or about 1.57 radians. The figure below shows decimal approximations for the quadrantal angles.
| Degrees | Radians: Exact Values |
Radians: Decimal Approximations |
| [latex]0°[/latex] | [latex]0[/latex] | [latex]0[/latex] |
| [latex]90°[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]1.57[/latex] |
| [latex]180°[/latex] | [latex]\pi[/latex] | [latex]3.14[/latex] |
| [latex]270°[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]4.71[/latex] |
| [latex]360°[/latex] | [latex]2\pi[/latex] | [latex]6.28[/latex] |
Note 6.5.
Because they are “benchmarks” for comparing angles, you should be very familiar with both the exact values of these angles in radians and their approximations!
Example 6.6.
Solution
Look at the figure above. The second quadrant includes angles between [latex]\dfrac{\pi}{2}[/latex] and [latex]\pi{,}[/latex] 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 [latex]\dfrac{3\pi}{2}[/latex] and [latex]2\pi[/latex] radians, so it lies in the fourth quadrant.
Checkpoint 6.7.
Solution
It turns out that measuring angles in radians is useful for many applications besides calculating arclengths, so we need to start thinking in radians. To help that process, we’ll first learn to convert between degrees and radians.
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 [latex]2\pi[/latex] radians or to [latex]360°{,}[/latex] so
[latex]360° = 2\pi ~{radians}[/latex]
If we divide both sides of this equation by [latex]360{,}[/latex] we get a fraction that is equal to 1:
[latex]1° = \dfrac{2\pi ~{radians}}{360} = \dfrac{\pi ~{radians}}{180}[/latex]
And of course it is also true that
[latex]1 = \dfrac{180°}{\pi ~{radians}}[/latex]
Because multiplying by 1 does not change the value of a number, we can use these fractions to convert between degrees and radians.
Converting between degrees and radians.
- To convert from radians to degrees, we multiply the radian measure by [latex]\dfrac{180°}{\pi ~{radians}}{.}[/latex]
- To convert from degrees to radians, we multiply the degree measure by [latex]\dfrac{\pi ~{radians}}{180 }{.}[/latex]
Example 6.8.
- Convert 3 radians to degrees.
- Convert 3 degrees to radians.
Solution
- [latex]\displaystyle (3 ~{radians}) \times \left(\dfrac{180°}{\pi}\right) = \dfrac{540°}{\pi} \approx 171.97°[/latex]
- [latex]\displaystyle (3°) \times \left(\dfrac{\pi}{180}\right) = \dfrac{\pi}{60}\approx 0.0523~ {radians.}[/latex]
Checkpoint 6.9.
- Convert [latex]60°[/latex] to radians. Give both an exact answer and an approximation to three decimal places.
- Convert [latex]\dfrac{3\pi}{4}[/latex] radians to degrees.
Solution
- [latex]\dfrac{\pi}{3} \approx 1.047[/latex] radians
- [latex]\displaystyle 135°[/latex]
Thus, the fraction [latex]\dfrac{180°}{\pi}[/latex] (or its reciprocal [latex]\dfrac{\pi}{180 }[/latex]) gives us a conversion factor between degrees and radians:
Unit Conversion for Angles.
[latex]{\dfrac{180°}{\pi~{radians}} = 1}[/latex]
Note 6.10.
You can review the use of conversion factors in the Algebra Refresher at the end of this section.
From our conversion factor, we also learn that
[latex]{ 1~{radian} = \dfrac{180°}{\pi} \approx 57.325°}[/latex]
So while [latex]1°[/latex] is a relatively small angle, 1 radian is much larger — nearly [latex]60°{,}[/latex] 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.
Arclength Formula
Now let us return to our calculation of arclength, and we’ll see the first instance in which measuring angles in radians is useful: To calculate an arclength, we need only multiply the radius of the circle by the radian measure of the spanning angle, [latex]\theta{.}[/latex] Look again at our formula for arclength:
[latex]\text{Arclength}~ = \bigg( \text{ fraction of one revolution} \times 2\pi \bigg) \cdot r[/latex]
The quantity in parentheses, fraction of one revolution [latex]\times 2\pi{,}[/latex] is just the measure of the spanning angle in radians. Thus, if [latex]\theta[/latex] is measured in radians, we have the following simple formula for arclength, [latex]s{.}[/latex]
Arclength Formula.
On a circle of radius [latex]r{,}[/latex] the length [latex]s[/latex] of an arc spanned by an angle [latex]\theta[/latex] in radians is
[latex]{s = r\theta}[/latex]
Thus, there is a special relationship between arclength and radians. An angle of 1 radian spans an arc on a circle equal to the radius of the circle, as shown at right. And the length of any arc is just the measure of its spanning angle in radians times the radius of the circle.
So for instance, we can use the formula to find the arclength spanned by an angle of 2.5 radians on a circle of radius six inches:
[latex]s = r \theta = 6 (2.5) = 15 ~ {inches}[/latex]
We can also use the formula in the form [latex]\theta = \dfrac{s}{r}[/latex] to find an angle that spans a given arc. For example, an arclength equal to one radius determines a central angle of one radian, or about [latex]57.3°{.}[/latex] In the next example, we compute a change in latitude on the Earth’s surface.
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? (Latitude is measured in degrees north or south of the equator.)
Solution
We think of the distance 500 miles as an arclength on the surface of the Earth, as shown at right. Substituting [latex]s = 500[/latex] and [latex]r = 3960[/latex] into the arclength formula gives
[latex]500 = 3960 ~ \theta\\ \theta = \dfrac{500}{3960} = 0.1263~ {radians}[/latex]
To convert the angle measure to degrees, we multiply by [latex]\dfrac{180°}{\pi}[/latex] to get
[latex]0.1263\left(\dfrac{180°}{\pi}\right) \approx 7.238°[/latex]
Your latitude has changed by about [latex]7.238°{.}[/latex]
Checkpoint 6.12.
The distance around the face of a large clock from 2 to 3 is five feet. What is the radius of the clock?
Solution
[latex]5=r~\dfrac{2\pi}{12}, r=9.554[/latex] ft
Unit Circle
In the rest of this chapter, we will see how to use the trigonometric functions sine, cosine, and tangent when the input variable is measured in radians instead of degrees, and why making that change greatly increases the utility of those functions. In Section 4.1 we connected the sine and cosine to the coordinates of points on a unit circle, a circle of radius 1. Here is an important observation that will inform our study:
On a unit circle, [latex]r = 1{,}[/latex] so the arclength formula becomes [latex]s = \theta{.}[/latex] Thus, on a unit circle, an arc of length 1 determines a central angle of 1 radian, or about [latex]57.3 °{.}[/latex] And the measure of a (positive) angle in radians is equal to the length of the arc it spans. In other words, if we walk around the circle, each time we travel over an arclength of one radius, we turn through an angle of one radian.
Example 6.13.
Solution
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.
Solution
[latex]s=1 \cdot \dfrac{7\pi}{6} \approx 3.66[/latex] ft
Section 6.1 Summary
Vocabulary
- Arclength
- Radian
- Conversion factor
- Latitude
- Unit circle
Concepts
- The distance we travel around a circle of radius is proportional to the angle of displacement.
[latex]{Arclength}~ = ~ ({fraction ~~of ~~one ~~revolution}) \cdot (2\pi r)[/latex] - We measure angles in radians when we work with arclength.
Radians.
The radian measure of an angle is given by
[latex]({fraction ~~of ~~one ~~revolution}\times 2\pi)[/latex]
- An arclength equal to one radius determines a central angle of one radian.
- Radian measure can be expressed as multiples of [latex]\pi[/latex] or as decimals.
Degrees [latex]\dfrac{{Radians:}}{{Exact Values}}[/latex] [latex]\dfrac{{Radians: Decimal}}{{Approximations}}[/latex] [latex]0°[/latex] [latex]0[/latex] [latex]0[/latex] [latex]90°[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]1.57[/latex] [latex]180°[/latex] [latex]\pi[/latex] [latex]3.14[/latex] [latex]270°[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]4.71[/latex] [latex]360°[/latex] [latex]2\pi[/latex] [latex]6.28[/latex] - We multiply by the appropriate conversion factor to convert between degrees and radians.
Unit Conversion for Angles.
[latex]\dfrac{180°}{\pi~{radians}} = 1[/latex]
To convert from radians to degrees we multiply the radian measure by [latex]\dfrac{180°}{\pi}{.}[/latex]
To convert from degrees to radians we multiply the degree measure by [latex]\dfrac{\pi}{180}{.}[/latex]
-
Arclength Formula.
On a circle of radius [latex]r{,}[/latex] the length [latex]s[/latex] of an arc spanned by an angle [latex]\theta[/latex] in radians is
[latex]s = r\theta[/latex]
- On a unit circle, the measure of a (positive) angle in radians is equal to the length of the arc it spans.
Study Questions
- The length of a circular arc depends on what two variables?
- Define the radian measure of an angle.
- What is the conversion factor from radians to degrees?
- On a unit circle, the length of an arc is equal to what other quantity?