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# Arc Length

## Portion of a circle's circumference.

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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. After completing this Concept, you'll be able to answer these questions.

### Guidance

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 d\begin{align*}d\end{align*} is the diameter or r\begin{align*}r\end{align*} is the radius, the length of ABˆ=mABˆ360πd\begin{align*}\widehat{AB}=\frac{m \widehat{AB}}{360^\circ} \cdot \pi d\end{align*} or mABˆ3602πr\begin{align*}\frac{m \widehat{AB}}{360^\circ} \cdot 2 \pi r\end{align*}.

#### Example A

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

In the picture, the central angle that corresponds with PQˆ\begin{align*}\widehat{PQ}\end{align*} is 60\begin{align*}60^\circ\end{align*}. This means that mPQˆ=60\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 360\begin{align*}360^\circ\end{align*} in a circle, so 60\begin{align*}60^\circ\end{align*} would be 16\begin{align*}\frac{1}{6}\end{align*} of that (60360=16)\begin{align*}\left( \frac{60^\circ}{360^\circ}= \frac{1}{6} \right)\end{align*}. Therefore, the length of PQˆ\begin{align*}\widehat{PQ}\end{align*} is 16\begin{align*}\frac{1}{6}\end{align*} of the circumference.

length of PQˆ=162π(9)=3π

#### Example B

The arc length of ABˆ=6π\begin{align*}\widehat{AB} = 6 \pi\end{align*} and is 14\begin{align*}\frac{1}{4}\end{align*} the circumference. Find the radius of the circle.

If 6π\begin{align*}6 \pi\end{align*} is 14\begin{align*}\frac{1}{4}\end{align*} the circumference, then the total circumference is 4(6π)=24π\begin{align*}4(6 \pi )=24 \pi\end{align*}. To find the radius, plug this into the circumference formula and solve for r\begin{align*}r\end{align*}.

24π12=2πr=r

#### Example C

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

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

15π15150=mPQˆ3602π(18)=mPQˆ10=mPQˆ

Watch this video for help with the Examples above.

#### Concept Problem Revisited

In the picture below, the top piece of pizza is if it is cut into 8 pieces. Therefore, for 18\begin{align*}\frac{1}{8}\end{align*} of the pizza, one piece would have 4485.5 inches\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 110\begin{align*}\frac{1}{10}\end{align*} of the crust, one piece would have 44104.4 inches\begin{align*}\frac{44}{10} \approx 4.4 \ inches\end{align*} of crust.

### Vocabulary

Circumference is the distance around a circle. Arc length is the length of an arc or a portion of a circle’s circumference.

### Guided Practice

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

1.

2.

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.

1. Use the Arc Length formula.

PQˆPQˆPQˆ=1353602π(12)=3824π=9π

2. Use the Arc Length formula.

PQˆPQˆPQˆ=3602603602π(144)=518288π=80π

3. The entire length of the crust, or the circumference of the pizza, is 20π62.83 in\begin{align*}20 \pi \approx 62.83\ in\end{align*}. In 112\begin{align*}\frac{1}{12}\end{align*} of the pizza, one piece would have 62.83125.24\begin{align*}\frac{62.83}{12} \approx 5.24\end{align*} inches of crust.

### Practice

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

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

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

1. 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, $12.00 1. What is the crust (in) to price ($) ratio for The Little Cheese?
2. What is the crust (in) to price ($) ratio for The Little Cheese? 3. What is the crust (in) to price ($) ratio for The Little Cheese?
4. 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?

### Vocabulary Language: English

Arc

Arc

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

arc length

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

Circumference

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

Dilation

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

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

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.