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# 10.4: Circumference and Arc Length

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Find the circumference of a circle.
• Define the length of an arc and find arc length.

## Review Queue

1. Find a central angle in that intercepts \begin{align*}\widehat{CE}\end{align*}
2. Find an inscribed angle that intercepts \begin{align*}\widehat{CE}\end{align*}.
3. How many degrees are in a circle? Find \begin{align*}m \widehat{ECD}\end{align*}.
4. If \begin{align*}m \widehat{CE} =26^\circ\end{align*}, find \begin{align*}m \widehat{CD}\end{align*} and \begin{align*}m \angle CBE\end{align*}.

Know What? 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.

## Circumference of a Circle

Circumference: The distance around a circle.

The circumference can also be called the perimeter of a circle. However, we use the term circumference for circles because they are round. In order to find the circumference of a circle, we need to explore \begin{align*}\pi\end{align*} (pi).

Investigation 10-1: Finding \begin{align*}\pi\end{align*} (pi)

Tools Needed: paper, pencil, compass, ruler, string, and scissors

1. Draw three circles with radii of 2 in, 3 in, and 4 in. Label the centers of each \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*}.
2. Draw in the diameters and determine their lengths.
3. Take the string and outline each circle with it. Cut the string so that it perfectly outlines the circle. Then, lay it out straight and measure it in inches. Round your answer to the nearest \begin{align*}\frac{1}{8}\end{align*}-inch. Repeat this for the other two circles.
4. Find \begin{align*}\frac{circumference}{diameter}\end{align*} for each circle. Record your answers to the nearest thousandth.

You should see that \begin{align*}\frac{circumference}{diameter}\end{align*} approaches 3.14159... We call this number \begin{align*}\pi\end{align*}, the Greek letter “pi.” When finding the circumference and area of circles, we must use \begin{align*}\pi\end{align*}.

\begin{align*}\pi\end{align*}, or “pi”: The ratio of the circumference of a circle to its diameter. It is approximately equal to 3.14159265358979323846...

To see more digits of \begin{align*}\pi\end{align*}, go to http://www.eveandersson.com/pi/digits/.

From Investigation 10-1, we found that \begin{align*}\frac{circumference}{diameter}=\pi\end{align*}. Let’s solve for the circumference, \begin{align*}C\end{align*}.

\begin{align*}\frac{C}{d} &= \pi\\ C &= \pi d\end{align*}

We can also say \begin{align*}C=2 \pi r\end{align*} because \begin{align*}d=2r\end{align*}.

Circumference Formula: \begin{align*}C=\pi d\end{align*} or \begin{align*}C=2 \pi r\end{align*}

\begin{align*}d=2r\end{align*}

Example 1: Find the circumference of a circle with a radius of 7 cm.

Solution: Plug the radius into the formula.

\begin{align*}C=2 \pi (7)=14 \pi \approx 44 \ cm\end{align*}

Example 2: The circumference of a circle is \begin{align*}64 \pi\end{align*}. Find the diameter.

Solution: Again, you can plug in what you know into the circumference formula and solve for \begin{align*}d\end{align*}.

\begin{align*}64 \pi &= \pi d\\ 64 &= d\end{align*}

Example 3: A circle is inscribed in a square with 10 in. sides. What is the circumference of the circle? Leave your answer in terms of \begin{align*}\pi\end{align*}.

Solution: From the picture, we can see that the diameter of the circle is equal to the length of a side. \begin{align*}C=10 \pi \ in\end{align*}.

Example 4: Find the perimeter of the square. Is it more or less than the circumference of the circle? Why?

Solution: The perimeter is \begin{align*}P=4(10)=40 \ in\end{align*}. In order to compare the perimeter with the circumference we should change the circumference into a decimal.

\begin{align*}C=10 \pi \approx 31.42 \ in\end{align*}. This is less than the perimeter of the square, which makes sense because the circle is inside the square.

## Arc Length

In Chapter 9, we measured arcs in degrees. This was called the “arc measure” or “degree measure.” Arcs can also be measured in length, as a portion of the circumference.

Arc Length: The length of an arc or a portion of a circle’s circumference.

The arc length is directly related to the degree arc measure.

Example 5: Find the length of \begin{align*}\widehat{PQ}\end{align*}. Leave your answer in terms of \begin{align*}\pi\end{align*}.

Solution: 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*}

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*}.

Another way to write this could be \begin{align*}\frac{x^\circ}{360^\circ} \cdot 2 \pi r\end{align*}, where \begin{align*}x\end{align*} is the central angle.

Example 6: 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.

Solution: 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*}

Example 7: Find the measure of the central angle or \begin{align*}\widehat{PQ}\end{align*}.

Solution: 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*}

Example 8: The tires on a compact car are 18 inches in diameter. How far does the car travel after the tires turn once? How far does the car travel after 2500 rotations of the tires?

Solution: One turn of the tire is the circumference. This would be \begin{align*}C=18 \pi \approx 56.55 \ in\end{align*}. 2500 rotations would be \begin{align*}2500 \cdot 56.55 \ in=141371.67 \ in\end{align*}, 11781 ft, or 2.23 miles.

Know What? Revisited 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.

## Review Questions

• Questions 1-10 are similar to Examples 1 and 2.
• Questions 11-14 are similar to Examples 3 and 4.
• Questions 15-20 are similar to Example 5.
• Questions 21-23 are similar to Example 6.
• Questions 24-26 are similar to Example 7.
• Questions 27-30 are similar to Example 8.

Fill in the following table. Leave all answers in terms of \begin{align*}\pi\end{align*}.

1. 15
2. 4
3. 6
4. \begin{align*}84 \pi\end{align*}
5. 9
6. \begin{align*}25\pi\end{align*}
7. \begin{align*}2\pi\end{align*}
8. 36
1. Find the radius of circle with circumference 88 in.
2. Find the circumference of a circle with \begin{align*}d=\frac{20}{\pi} \ cm\end{align*}.

Square \begin{align*}PQSR\end{align*} is inscribed in \begin{align*}\bigodot T\end{align*}. \begin{align*}RS=8 \sqrt{2}\end{align*}.

1. Find the length of the diameter of \begin{align*}\bigodot T\end{align*}.
2. How does the diameter relate to \begin{align*}PQSR\end{align*}?
3. Find the perimeter of \begin{align*}PQSR\end{align*}.
4. Find the circumference of \begin{align*}\bigodot T\end{align*}.

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.

For questions 27-30, a truck has tires with a 26 in diameter.

1. How far does the truck travel every time a tire turns exactly once? What is this the same as?
2. How many times will the tire turn after the truck travels 1 mile? (1 mile = 5280 feet)
3. The truck has travelled 4072 tire rotations. How many miles is this?
4. The average recommendation for the life of a tire is 30,000 miles. How many rotations is this?

1. \begin{align*}\angle CAE\end{align*}
2. \begin{align*}\angle CBE\end{align*}
3. \begin{align*}360^\circ, 180^\circ\end{align*}
4. \begin{align*}m\widehat{CD} = 180^\circ - 26^\circ = 154^\circ, m \angle CBE = 13^\circ\end{align*}

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