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 $45°$ is ${\displaystyle \frac{1}{8}}$ of a complete revolution, so the the length of the arc from point $A$ to point $B,$ called $s$ in the figure at right, is ${\displaystyle \frac{1}{8}}$ of the circumference. Thus

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

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

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

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

The Ferris wheel in the introduction has circumference

$C\approx 2\pi (100)=628feet$

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

$s=0,s\approx 157,s\approx 314,ands\approx 471,$

as shown at right.

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

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

to

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

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

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

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 ${\displaystyle \frac{1}{4}}(2\pi )={\displaystyle \frac{\pi }{2}},$ or about 1.57 radians. The figure below shows decimal approximations for the quadrantal angles.

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 $2\pi$ radians or to $360°,$ so

$360°=2\pi radians$

If we divide both sides of this equation by $360,$ we get a fraction that is equal to 1:

$1°={\displaystyle \frac{2\pi radians}{360}}={\displaystyle \frac{\pi radians}{180}}$

And of course it is also true that

$1={\displaystyle \frac{180°}{\pi radians}}$

Because multiplying by 1 does not change the value of a number, we can use these fractions to convert between degrees and radians.

Thus, the fraction ${\displaystyle \frac{180°}{\pi }}$ (or its reciprocal ${\displaystyle \frac{\pi }{180}}$) gives us a conversion factor between degrees and radians:

From our conversion factor, we also learn that

$1radian={\displaystyle \frac{180°}{\pi }}\approx 57.325°$

So while $1°$ is a relatively small angle, 1 radian is much larger — nearly $60°,$ 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, $\theta .$ Look again at our formula for arclength:

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

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 simple formula for arclength, $s.$

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:

$s=r\theta =6(2.5)=15inches$

We can also use the formula in the form $\theta ={\displaystyle \frac{s}{r}}$ 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 $57.3°.$ In the next example, we compute a change in latitude on the Earth’s surface.

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, $r=1,$ so the arclength formula becomes $s=\theta .$ Thus, on a unit circle, an arc of length 1 determines a central angle of 1 radian, or about $57.3°.$ 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.

An ant walks around the rim of a circular birdbath of diameter 2 feet. How far has the ant walked when it has turned through an angle of $210°?$

Solution

$s=1\cdot {\displaystyle \frac{7\pi }{6}}\approx 3.66$ ft

Section 6.1 Summary

Vocabulary

• Arclength
• Radian
• Conversion factor
• Latitude
• Unit circle