Have you ever seen a discus? Take a look at this dilemma.

“I don’t know how to figure this out,” Jesse said to his friend Emory one morning.

“Figure what out?” Emory inquired.

“I have to figure out the distance around the discus ring. That is what Mrs. Henry asked me to figure out,” Jesse said.

“Well, what do you know?”

“I know that the shape of it is a circle. I also know that the diameter of the circle is 8 feet. I need the circumference of the ring now and that is where I am stuck,” Jesse explained.

“That’s not so hard,” Emory said.

Jesse looked at his friend puzzled.

**Do you know what Emory knows? In this Concept you will learn all about circles. At the end of the Concept, you will see this problem again. Then you will need to help Jesse solve for the area of the discus ring.**

### Guidance

** Circles** are unique geometric figures.

**A circle is the set of points that are equidistant from a center point.**

**The** *radius***of a circle is the distance from the center to any point on the circle. The diameter is the distance across the circle through the center. The** *diameter***is always twice as long as the radius.**

**We also use the special number** *pi***when dealing with circle calculations.**

*Pi* is a decimal that is infinitely long (3.14159265...), but in our calculations we round it to 3.14.

We use the symbol \begin{align*}\pi\end{align*} to represent this number.

Pi is the ratio of the ** circumference**, or

**distance around a circle**, to the diameter.

In other words, these two measurements are related. If we change the diameter, the circumference changes proportionally. For example, if we double the length of the diameter, the circumference doubles also.

**Let’s review working with the radius and diameter while finding the circumference.**

As the diameter of the circle grows, the circumference of the circle grows at the same rate.

In other words, however the diameter of the circle changes, the circumference of the circle must change exactly the same way. This is a proportional relationship.

We express this proportional relationship as a ratio.

**A ratio simply means that two numbers are related to each other.**

Circles are special in geometry because this ratio of the circumference and the diameter always stays the same.

We can see this when we divide the circumference of a circle by its diameter. No matter how big or small the circle is, we will always get the same number.

Let’s try it out on the circles below.

\begin{align*}\frac{Circumference}{Diameter} &=\frac{6.28}{2}=3.14\\ \frac{Circumference}{Diameter} &=\frac{12.56}{4}=3.14\end{align*}

**Even though we have two different circles, the result is the same. Therefore the circumference and the diameter always exist in equal proportion, or a ratio, with each other. Whenever we divide the circumference by the diameter, we will always get 3.14,** *pi***.**

Using the equations above, we can write a general formula that shows the relationship between *pi*, circumference, and diameter. When we rearrange it, we get the formula for the circumference of a circle.

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

If we divide the circumference by the diameter to find pi, then we can use the formula circumference equals pi times the diameter to find the circumference of any circle.

Take a look at this one.

**What is the circumference of a circle that has a diameter of 3 inches?**

To find the circumference, we can substitute these values into the formula.

\begin{align*}C &=\pi(3)\\ C &=3.14(3)\\ C &=9.42 \ inches \end{align*}

**What about if we were given the measurement for the radius instead of the diameter?**

Well, we know that the radius is one – half of the diameter, so we can use the following formula or you can figure out the measurement for the diameter by using mental math.

\begin{align*}C=2 \pi r\end{align*}

**You can see that we can use either the measurement for the radius or for the diameter to find the measurement for the circumference.**

*Write both of these formulas down in your notebook.*

Find the circumference of each circle given the diameter or radius.

#### Example A

Diameter = 6 inches

**Solution: 18.84 inches**

#### Example B

Radius = 4.5 feet

**Solution: 28.26 feet**

#### Example C

Diameter = 3.5 meters

**Solution: 10.99 or 11 meters**

Now let's go back to the dilemma from the beginning of the Concept.

The diameter of the discus ring is 8 feet. We can use the following formula to figure out the circumference of the ring.

\begin{align*}C &=\pi(8)\\ C &=3.14(8)\\ C &=25.12 \ feet \end{align*}

**This is our answer.**

### Vocabulary

- Circle
- all points are equidistant from a center point.

- Radius
- the distance half-way across a circle.

- Diameter
- the distance across a circle.

- Circumference
- the distance around a circle.

### Guided Practice

Here is one for you to try on your own.

**What is the circumference of a circle if the radius is 2.5 feet?**

**Solution**

First, we can find the diameter using this measurement. If the radius is 2.5 feet, then the diameter is 5 feet. Let’s find the circumference using this measurement.

\begin{align*} C &=3.14(5)\\ C &=15.7 \ feet\end{align*}

**We could also have used the radius alone to find the circumference. We just use a different formula.**

\begin{align*}C &=2(3.14)(2.5)\\ C &=15.7 \ feet\end{align*}

### Video Review

### Practice

Directions: Find the circumference of each circle given the radius or diameter.

- \begin{align*}d=10 \ in\end{align*}
- \begin{align*}d=5 \ in\end{align*}
- \begin{align*}d=7 \ ft\end{align*}
- \begin{align*}d=12 \ mm\end{align*}
- \begin{align*}d=14 \ cm\end{align*}
- \begin{align*}r=4 \ in\end{align*}
- \begin{align*}r=6 \ meters\end{align*}
- \begin{align*}r=8 \ ft.\end{align*}
- \begin{align*}r=11 \ in\end{align*}
- \begin{align*}r=15 \ cm\end{align*}

Directions: Find the diameter given each circumference.

11. \begin{align*}53.38 \ inches\end{align*}

12. \begin{align*}43.96 \ feet\end{align*}

13. \begin{align*}56.52 \ inches\end{align*}

14. \begin{align*}65.94 \ meters\end{align*}

15. \begin{align*}48.67 \ meters\end{align*}

16. \begin{align*}37.68 \ feet\end{align*}

17. \begin{align*}78.5 \ meters\end{align*}

18. \begin{align*}100.48 \ cm\end{align*}