# 8.4: Arc Length

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

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

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

The circumference of the circle is: \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 \begin{align*}\frac{1}{6}\end{align*} of the entire circumference of the circle.

The **arc length** of \begin{align*}\widehat{PQ}\end{align*} is: \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 \begin{align*}C\end{align*}
- diameter \begin{align*}d\end{align*}
- radius \begin{align*}r\end{align*}

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

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

**Arc length**

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

**Reading Check:**

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

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

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

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

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?*

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

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

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

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

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

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

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

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

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

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