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# Circumference

## Pi times the diameter or the distance around the circle.

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Circumference

### Circumference

Circumference is 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.

Circumference Formula: C=πd\begin{align*}C=\pi d\end{align*} where the diameter d=2r\begin{align*}d=2r\end{align*}, or twice the radius. So C=2πr\begin{align*}C=2 \pi r\end{align*} as well.

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

You should have a π\begin{align*}\pi\end{align*} button on your calculator. If you don't, you can use 3.14 as an approximation for π\begin{align*}\pi\end{align*}. You can also leave your answers in terms of π\begin{align*}\pi\end{align*} for many problems.

What if you were given the radius or diameter of a circle? How could you find the distance around that circle?

### Examples

#### Example 1

Find the circumference of a circle with a radius of 7 cm.

Plug the radius into the formula.

C=2π(7)=14π44 cm\begin{align*}C=2 \pi (7)=14 \pi \approx 44 \ cm\end{align*}

#### Example 2

The circumference of a circle is 64π\begin{align*}64 \pi\end{align*} units. Find the diameter.

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

64π64 units=πd=d\begin{align*}64 \pi &= \pi d\\ 64 \ units &= 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 π\begin{align*}\pi\end{align*}.

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

#### Example 4

Find the perimeter of the square in Example 3. Is it more or less than the circumference of the circle? Why?

The perimeter is P=4(10)=40 in\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.

C=10π31.42 in\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.

#### Example 5

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?

One turn of the tire is the circumference. This would be C=18π56.55 in\begin{align*}C=18 \pi \approx 56.55 \ in\end{align*}. 2500 rotations would be 250056.55 in approx141,375 in\begin{align*}2500 \cdot 56.55 \ in \ approx 141,375 \ in\end{align*}, 11,781 ft, or 2.23 miles.

### Review

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

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

Square PQSR\begin{align*}PQSR\end{align*} is inscribed in T\begin{align*}\bigodot T\end{align*}. RS=82\begin{align*}RS=8 \sqrt{2}\end{align*}.

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

For questions 14-17, 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?

To see the Review answers, open this PDF file and look for section 10.8.

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### Vocabulary Language: English Spanish

TermDefinition
chord A line segment whose endpoints are on a circle.
circle The set of all points that are the same distance away from a specific point, called the center.
Circumference The circumference of a circle is the measure of the distance around the outside edge of a circle.
diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
pi (or $\pi$) The ratio of the circumference of a circle to its diameter.
Radius The radius of a circle is the distance from the center of the circle to the edge of the circle.
Congruent Congruent figures are identical in size, shape and measure.
Regular Polygon A regular polygon is a polygon with all sides the same length and all angles the same measure.

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