# 10.9: Arc Length

**At Grade**Created by: CK-12

**Practice**Arc Length

What if you wanted to find the "length" of the crust for an individual slice of pizza? A typical large pizza has a diameter of 14 inches and is cut into 8 or 10 pieces. If the "length" of the entire crust is the circumference of the pizza, find the "length" of the crust for one piece of pizza when the entire pizza is cut into a) 8 pieces or b) 10 pieces.

### Arc Length

One way to measure arcs is in degrees. This is called the “arc measure” or “degree measure.” 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:** If \begin{align*}d\end{align*} is the diameter or \begin{align*}r\end{align*} is the radius, 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*}.

#### Measuring Arc Length

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*} as well. So, 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.

\begin{align*}length \ of \ \widehat{PQ} =\frac{1}{6} \cdot 2 \pi (9)=3 \pi\end{align*}

#### Finding the Radius

The arc length of \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 &= r\end{align*}

#### Measuring the Central Angle

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*}

#### Pizza Problem Revisited

In the picture below, the top piece of pizza is if it is cut into 8 pieces. Therefore, for \begin{align*}\frac{1}{8}\end{align*} of the pizza, one piece would have \begin{align*}\frac{44}{8} \approx 5.5 \ inches\end{align*} of crust. The bottom piece of pizza is if the pizza is cut into 10 pieces. For \begin{align*}\frac{1}{10}\end{align*} of the crust, one piece would have \begin{align*}\frac{44}{10} \approx 4.4 \ inches\end{align*} of crust.

### Examples

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*}.

#### Example 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*}

#### Example 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*}

#### Example 3

An extra large pizza has a diameter of 20 inches and is cut into 12 pieces. Find the length of the crust for one piece of pizza.

The entire length of the crust, or the circumference of the pizza, is \begin{align*}20 \pi \approx 62.83\ in\end{align*}. In \begin{align*}\frac{1}{12}\end{align*} of the pizza, one piece would have \begin{align*}\frac{62.83}{12} \approx 5.24\end{align*} inches of crust.

### Review

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.

- The Olympics symbol is five congruent circles arranged as shown below. Assume the top three circles are tangent to each other. Brad is tracing the entire symbol for a poster. How far will his pen point travel?

Mario’s Pizza Palace offers a stuffed crust pizza in three sizes (diameter length) for the indicated

prices:

The Little Cheese, 8 in, $7.00

The Big Cheese, 10 in, $9.00

The Cheese Monster, 12 in, $12.00

- What is the crust (in) to price ($) ratio for The Little Cheese?
- What is the crust (in) to price ($) ratio for The Little Cheese?
- What is the crust (in) to price ($) ratio for The Little Cheese?
- Michael thinks the cheesy crust is the best part of the pizza and wants to get the most crust for his money. Which pizza should he buy?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 10.9.

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Arc

An arc is a section of the circumference of a circle.arc length

In calculus, arc length is the length of a plane function curve over an interval.Circumference

The circumference of a circle is the measure of the distance around the outside edge of a circle.Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.radian

A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.Sector

A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees.### Image Attributions

Here you'll learn how to find the length of an arc.

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