What if you were given the angle measure of a circle's arc? How could you find the length of that arc? After completing this Concept, you'll be able to find an arc's length, or its portion of a circle's circumference.

### Watch This

### Guidance

One way to measure arcs is in degrees. This is called the “arc measure” or “degree measure” (see Arcs in Circles). Arcs can also be measured in length, as a portion of the circumference. **Arc length** is the length of an arc or a portion of a circle’s circumference. The arc length is directly related to the degree arc measure.

**Arc Length Formula:** The length of \begin{align*}\widehat{AB}=\frac{m \widehat{AB}}{360^\circ} \cdot \pi d\end{align*} or \begin{align*}\frac{m \widehat{AB}}{360^\circ} \cdot 2 \pi r\end{align*}.

#### Example A

Find the length of \begin{align*}\widehat{PQ}\end{align*}. Leave your answer in terms of \begin{align*}\pi\end{align*}.

In the picture, the central angle that corresponds with \begin{align*}\widehat{PQ}\end{align*} is \begin{align*}60^\circ\end{align*}. This means that \begin{align*}m \widehat{PQ}=60^\circ\end{align*}. Think of the arc length as a portion of the circumference. There are \begin{align*}360^\circ\end{align*} in a circle, so \begin{align*}60^\circ\end{align*} would be \begin{align*}\frac{1}{6}\end{align*} of that \begin{align*}\left(\frac{60^\circ}{360^\circ}=\frac{1}{6}\right)\end{align*}. Therefore, the length of \begin{align*}\widehat{PQ}\end{align*} is \begin{align*}\frac{1}{6}\end{align*} of the circumference. *length of* \begin{align*}\widehat{PQ}=\frac{1}{6} \cdot 2 \pi (9)=3 \pi\end{align*} units.

#### Example B

The arc length of a circle is \begin{align*}\widehat{AB}=6 \pi\end{align*} and is \begin{align*}\frac{1}{4}\end{align*} the circumference. Find the radius of the circle.

If \begin{align*}6 \pi\end{align*} is \begin{align*}\frac{1}{4}\end{align*} the circumference, then the total circumference is \begin{align*}4(6 \pi )=24 \pi\end{align*}. To find the radius, plug this into the circumference formula and solve for \begin{align*}r\end{align*}.

\begin{align*}24 \pi &= 2 \pi r\\ 12 \ units &= r\end{align*}

#### Example C

Find the measure of the central angle or \begin{align*}\widehat{PQ}\end{align*}.

Let’s plug in what we know to the Arc Length Formula.

\begin{align*}15 \pi &= \frac{m \widehat{PQ}}{360^\circ} \cdot 2 \pi (18)\\ 15 &= \frac{m \widehat{PQ}}{10^\circ}\\ 150^\circ &= m \widehat{PQ}\end{align*}

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### Guided Practice

Find the arc length of \begin{align*}\widehat{PQ}\end{align*} in \begin{align*}\bigodot A\end{align*}. Leave your answers in terms of \begin{align*}\pi\end{align*}.

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3. A typical large pizza has a diameter of 14 inches and is cut into 8 pieces. Think of the crust as the circumference of the pizza. Find the *length* of the crust for the entire pizza. Then, find the length of the crust for one piece of pizza if the entire pizza is cut into 8 pieces.

**Answers:**

1. Use the Arc Length formula.

\begin{align*}\widehat{PQ}&=\frac{135}{360}\cdot 2 \pi (12)\\ \widehat{PQ}&=\frac{3}{8}\cdot 24 \pi \\ \widehat{PQ}&=9\pi\end{align*}

2. Use the Arc Length formula.

\begin{align*}\widehat{PQ}&=\frac{360-260}{360}\cdot 2 \pi (144)\\ \widehat{PQ}&=\frac{5}{18}\cdot 288 \pi \\ \widehat{PQ}&=80\pi\end{align*}

3. The entire length of the crust, or the circumference of the pizza, is \begin{align*}14 \pi \approx 44 \ in\end{align*}. In \begin{align*}\frac{1}{8}\end{align*} of the pizza, one piece would have \begin{align*}\frac{44}{8} \approx 5.5\end{align*} inches of crust.

### Explore More

Find the arc length of \begin{align*}\widehat{PQ}\end{align*} in \begin{align*}\bigodot A\end{align*}. Leave your answers in terms of \begin{align*}\pi\end{align*}.

Find \begin{align*}PA\end{align*} (the radius) in \begin{align*}\bigodot A\end{align*}. Leave your answer in terms of \begin{align*}\pi\end{align*}.

Find the central angle or \begin{align*}m \widehat{PQ}\end{align*} in \begin{align*}\bigodot A\end{align*}. Round any decimal answers to the nearest tenth.