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# 8.4: Arc Length

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Calculate the length of an arc of a circle.

## Arc Length

Arcs are measured in two different ways:

Degree measure: The degree measure of an arc is the fractional part of a 360\begin{align*}360^\circ\end{align*} complete circle that the arc is.

Linear measure: This is the length, in units such as centimeters and feet, if you traveled from one end of the arc to the other end.

• Arcs can be measured in __________ ways, _____________________________ measure and ___________________________ measure.

Example 1

Find the length of PQˆ\begin{align*}\widehat{PQ}\end{align*} if mPQˆ=60\begin{align*}m \widehat{PQ} = 60^\circ\end{align*}. The radius of the circle is 9 inches.

Remember, 60\begin{align*}60^\circ\end{align*} is the measure of the central angle associated with mPQˆ\begin{align*}m \widehat{PQ}\end{align*}. This is the degree measure of the arc.

To find the linear measure of the arc, or mPQˆ\begin{align*}m \widehat{PQ}\end{align*}, we use the fact that it is 60360=16\begin{align*}\frac{60}{360} = \frac{1}{6}\end{align*} of an entire circle.

The circumference of the circle is: C=πd=2πr=2π(9)=18π inches\begin{align*}C = \pi d = 2 \pi r = 2 \pi (9) = 18 \pi \ inches\end{align*}

The length of the arc, in this case, is 16\begin{align*}\frac{1}{6}\end{align*} of the entire circumference of the circle.

The arc length of PQˆ\begin{align*}\widehat{PQ}\end{align*} is: 1618π=18π6=3π\begin{align*}\frac{1}{6} \cdot 18 \pi = \frac{18 \pi}{6} = 3 \pi\end{align*} inches or 9.42 inches

In this lesson we study the second type of arc measure—the linear measure of an arc, or the arc’s length. Arc length is directly related to the degree measure of an arc.

Suppose a circle has:

• circumference C\begin{align*}C\end{align*}
• diameter d\begin{align*}d\end{align*}
• radius r\begin{align*}r\end{align*}

Also, suppose an arc of the circle has degree measure m\begin{align*}m\end{align*}.

Realize that m360\begin{align*}\frac{m}{360}\end{align*} is the fractional part of the circle that the arc represents.

Arc length

Arc Length=m360C=m360πd=m3602πr\begin{align*}Arc \ Length = \frac{m}{360} \cdot C = \frac{m}{360} \cdot \pi d = \frac{m}{360} \cdot 2 \pi r\end{align*}

1. In your own words, describe the linear measure of an arc:

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2. True/False: The degree measure of an arc is exactly the same as the linear measure of an arc.

3. How could you correct the statement in #2 above to make it true?

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\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}{\;}\end{align*}

4. Why do we use the fraction m360\begin{align*}\frac{m}{360}\end{align*} to calculate arc length? Describe.

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\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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