Remember Miguel from the Find Areas of Combined Figures Involving Triangles Concept? Well, now he is going to find the circumference of the circle. Take a look.

Miguel’s latest task is to measure some different “on deck” pads for the pitchers to practice with. An on deck pad is a circular pad that is made up of a sponge and some fake grass. Pitcher’s practice their warm-ups by standing on them. They work on stretching and get ready to “pitch” the ball prior to their turn on the mound.

Miguel has three different on deck pads that he is working with. The coach has asked him to measure each one and find the circumference and the area of each.

Miguel knows that the circumference is the distance around the edge of the circle. He decides to start with figuring out the circumference of each circle.

He measures the distance across each one.

The first one measures 4 ft. across.

The second measures 5 ft. across.

The third one measures 6 ft across.

Miguel begins working on his calculations.

**Pay attention to how we can use the formula for finding circumference. Then you will be able to help Miguel at the end of the Concept.**

### Guidance

Circumference is a word associated with circles. **A** *circle***is a figure whose edge is made of points that are all the same distance from the center.** This lesson is all about the circumference of circles. Let’s begin by taking a look at what we mean when we use that word. **The** *circumference***of a circle is the distance around the outside edge of a circle.**

With other figures, we could find the perimeter of the circle. **The** *perimeter***is the distance around a polygon.** Circles are not polygons because they are not made up of line segments. When we were finding the perimeter of a polygon, we found the sum of the outer edges.

**Circles are quite different. We can’t add up the measurement of the edges, because there aren’t any. To understand circumference, we have to begin by looking at the parts of a circle.**

**What are the parts of a circle?**

You can see here that this is one part of a circle. **It is the distance from the center of the circle to the outside edge of the circle. This measurement is called the** *radius***.**

**You can see that the distance across the center of the circle is called the** *diameter***. The diameter divides the circle into two equal halves. It is twice as long as the radius.**

**Now we know the basics of circles: the radius, the diameter, and the circumference. Let’s see how we can use these elements in a formula to find the circumference of a circle.**

**Let’s think about the relationship between the diameter and the circumference.**

Think about circles that are drawn on a playground. There are many different sizes and shapes of them. If we were to draw one circle with chalk, that circle would have a diameter and a circumference. If we were to draw a circle around the outside of this other circle, it would have a longer diameter and therefore it would have a larger circumference.

**There is a relationship between the size of the diameter and the size of the circumference.**

**What is this relationship?**

**It is a proportional relationship that is expressed 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.

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 above.

\begin{align*}\frac{\text{circumference}}{\text{diameter}} \quad = \quad \frac{6.28}{2} \quad = \quad 3.14 && \frac{\text{circumference}}{\text{diameter}} \quad = \quad \frac{12.56}{4} \quad = \quad 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. This relationship is always the same. Whenever we divide the circumference by the diameter, we will always get 3.14. We call this number** *pi***, and we represent it with the symbol \begin{align*}\pi\end{align*}.** *Pi* is actually a decimal that is infinitely long—it has no end. We usually round it to 3.14 to make calculations easier.

**Using the equations above, we can write a general formula that shows the relationship between** *pi***, circumference, and diameter.**

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

**If we rearrange this formula, we can also use it to find the circumference of a circle when we are given the diameter.**

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

**We can use this formula to find the circumference of any circle. Remember, the number for \begin{align*}\pi\end{align*} is always the same: 3.14. We simply multiply it by the diameter to get the circumference.**

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

Now that you understand the parts of a circle and how the formula for finding the circumference is developed, it is time to put this formula into practice. Let’s look at using it to figure out the circumference of a circle.

Find the circumference of the circle below.

We can see that the diameter of the circle is 8 inches. Let’s put this number into the formula.

\begin{align*}C & = \pi d\\ C & = \pi (8)\\ C & = 25.12 \ in.\end{align*}

**The circumference of a circle that has a diameter of 8 inches is 25.12 inches. In other words, if we could unroll the circle into a flat line, it would be 25.12 inches long.**

Let’s try another.

What is the area of the circle below?

Again, we know the diameter, so we put it into the formula and solve.

\begin{align*}C & = \pi d\\ C & = \pi (12.7)\\ C & = 39.88 \ m\end{align*}

**We can round this number up to the nearest hundredth and say that a circle with a diameter of 12.7 meters has a circumference of approximately 39.88 meters.**

**What if we had been given the radius and not the diameter?**

Well, the radius is one-half as long as the diameter. So we can multiply the radius by 2 and end up with the same measure as the diameter. Here is how we can alter the formula when given a radius.

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

Let's try this out.

Find the circumference of the following circle.

Now let’s substitute the known information into the formula.

\begin{align*}C&=2 \pi (3) \\ C&=2(3.14)(3) \\ C&=3.14(6) \\ C&=18.84 \ inches \end{align*}

**This is our answer.**

Now it’s time for you to try a few on your own. Find the circumference of each circle given the radius or diameter.

#### Example A

\begin{align*}d = 5 \ in\end{align*}

**Solution: \begin{align*}15.7\end{align*} in**

#### Example B

\begin{align*}r = 3.5 \ in\end{align*}

**Solution: \begin{align*}10.99\end{align*} in**

#### Example C

\begin{align*}d = 10 \ ft\end{align*}

**Solution: \begin{align*}31.4\end{align*} inches**

Here is the original problem once again.

Miguel’s latest task is to measure some different “on deck” pads for the pitchers to practice with. An on deck pad is a circular pad that is made up of a sponge and some fake grass. Pitcher’s practice their warm-ups by standing on them. They work on stretching and get ready to “pitch” the ball prior to their turn on the mound.

Miguel has three different on deck pads that he is working with. The coach has asked him to measure each one and find the circumference and the area of each.

Miguel knows that the circumference is the distance around the edge of the circle. He decides to start with figuring out the circumference of each circle.

He measures the distance across each one.

The first one measures 4 ft. across.

The second measures 5 ft. across.

The third one measures 6 ft across.

Miguel begins working on his calculations.

**Let’s start with the first on deck pad. It measures 4 feet across. That is our diameter. We multiply that times 3.14 to find the circumference.**

\begin{align*}C&=\pi d\\ C&=3.14(4)\\ C&=12.56 \ ft\end{align*}

**Next, we work on the deck pad with a diameter of five feet.**

\begin{align*}C&=\pi d\\ C&=3.14(5)\\ C&=15.7 \ ft\end{align*}

**Finally we work on the deck pad with a diameter of 6 feet.**

\begin{align*}C&=\pi d\\ C&=3.14(6)\\ C&=18.84 \ ft\end{align*}

**Miguel jots down these dimensions. Now he is ready to figure out the area of each on deck pad.**

### Vocabulary

- Circumference
- the distance around the outside edge of a circle.

- Perimeter
- the distance around the edge of a polygon.

- Circle
- a series of connected points equidistant from a center point.

- Radius
- the distance halfway across a circle.

- Diameter
- the distance across the center of circle.

- Pi
- the ratio of circumference to diameter, 3.14.

- Ratio
- a comparison between two quantities.

### Guided Practice

Here is one for you to try on your own.

Samuel baked a pie in a 9-inch pie pan. What is the circumference of the pie?

**Answer**

Let’s begin by figuring out what the problem is asking us to find. We need to find the circumference of the pie, so we will use the formula to solve for \begin{align*}C\end{align*}. In order to use the formula, we need to know either the diameter or the radius of the pie. The problem tells us that the diameter of the pie is 9 inches. Let’s put this information into the formula and solve for the circumference.

\begin{align*}C & = \pi d\\ C & = \pi (9)\\ C & = 28.26 \ in.\end{align*}

**The circumference of the pie is 28.26 inches.**

### Video Review

- This is a James Sousa video on determining the circumference of a circle.

### Practice

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

1. radius = 2 in

2. diameter = 4 ft

3. radius = 4.5 in

4. diameter = 8 meters

5. radius = 12 inches

6. diameter = 12 mm

7. radius = 14 mm

8. diameter = 13 feet

9. radius = 10 inches

10. diameter = 7.5 feet

11. radius = 2.5 inches

12. radius = 5.6 feet

13. diameter = 3.75 feet

14. diameter = 4.5 feet

15. diameter = 10.75 meters