Arclength on a Circle.
[latex]\text{Arclength}~ = \bigg( \text{ fraction of one revolution} \bigg) \cdot \bigg (2\pi r \bigg)[/latex]
Chapter 6: Radians
Use the appropriate conversion factor to convert units.
1. [latex]\dfrac{1~ {mile}}{1.609~{kilometers}} = 1[/latex]
2. [latex]\dfrac{1~ {acre}}{0.405~{hectare}} = 1[/latex]
3. [latex]\dfrac{1~ {horsepower}}{746~{watts}} = 1[/latex]
4. [latex]\dfrac{1~ {troy ounce}}{480~{grains}} = 1[/latex]
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
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.
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.
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