# 10.5: Pi

**At Grade**Created by: CK-12

**Practice**Pi

Have you ever tried to find the right tablecloth to put on a table? Jillian is about to tackle just that task.

Jillian is trying to find the right table cloth for the round table where the quilters will work. She has measured the distance across the table.

"There has to be a connection between the distance across the table and the distance around it," she thinks to herself.

Is Jillian correct?

**In this Concept, you will learn about pi and this question will be answered.**

### Guidance

To work with circles, we first need to review the parts of a circle. Let’s begin there.

We can measure several key parts of a circle. We can measure the distance across the center of the circle. This distance is called the *diameter* of the circle.

Here is a picture of the diameter.

We can measure the distance from the center of the circle to the outer edge. This distance is called the *radius*. Notice that the radius is one-half of the measure of the diameter.

Here is a picture of the radius.

We can measure the perimeter of the circle too. This distance is called the *circumference* of the circle.

To understand things about circles, let’s travel back in time.

Here is our time machine.

We have traveled all the way back to a time when the Greeks were discovering all sorts of things about mathematics. They were puzzled by mathematics and by the relationships between different measurements and geometry. The Greeks were famous for investigating ratios and proportions. When they studied different things, they knew that there was a connection between shapes and their measurements. Some of the Greeks thought a lot about circles.

Although the Babylonians had been investigating circles too, it was a Greek man named Archimedes who is credited with figuring out that there is a relationship between the diameter and circumference of a circle.

Archimedes discovered that if you take the distance across the circle and stretch it around the circumference, that the length of the diameter will go around the circle 3 and a bit more times.

Let’s say that the diameter of this circle is 5 cm, in that case the circumference of the circle is three and a little more times the 5 cm, or a little less than 16 cm.

**We say that the ratio of the diameter to the circumference is pi. We use the number 3.14 for pi because the actual ratio is a non-terminating decimal, which means it cannot be written precisely as a numeral since the decimal places never end or form a pattern. However, using two decimal places for pi works for estimating the circumference of the circle.**

Here is the symbol for pi. When you see this symbol, you can use 3.14 in your arithmetic.

Think about what you have learned and answer these questions.

#### Example A

Who was the first person to figure out the relationship between the diameter and the circumference?

**Solution: Archimedes**

#### Example B

What is the distance across the circle called?

**Solution: The diameter**

#### Example C

What is the distance around the circle called?

**Solution: The circumference**

Now let's go back to Jillian and the question about the tablecloth.

Jillian is trying to find the right table cloth for the round table where the quilters will work. She has measured the distance across the table.

"There has to be a connection between the distance across the table and the distance around it," she thinks to herself.

Is Jillian correct?

Jillian is correct. The ratio of the diameter to the circumference is what she is thinking about. This important measurement called "pi" is indicated with the value 3.14. When we work with circles, it is important to always remember pi.

### Vocabulary

Here are the vocabulary words in this Concept.

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

- Diameter
- the measure of the distance across the center of a circle.

- Radius
- the measure of the distance half-way across the circle. It is the measure from the center to the outer edge. The radius is also half the length of the diameter.

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

- Archimedes
- a Greek mathematician and philosopher who identified 3.14 as pi.

### Guided Practice

Here is one for you to try on your own.

If the radius of a circle is 3, what is the diameter?

**Answer**

To answer this question, you must know the definitions of radius and diameter. The radius is one - half the diameter of a circle.

**The answer is 6.**

### Video Review

Here is a video for review.

Khan Academy: Circles: Radius, Diameter and Circumference

### Practice

Directions: Given each radius, identify the diameter.

1. r = 4 in.

2. r = 6 in

3. r = 5 in

4. r = 12 in

5. r = 16 ft.

6. r = 28 mm.

7. r = 12.5 ft.

8. r = 1.25 m

Directions: Given each diameter, identify the radius.

9. d = 12 m

10. d = 18 m

11. d = 12.5 in

12. d = 18.5 ft

13. d = 9.8 in

14. d = 1.45 mm

15. d = 1.75 ft

16. d = 2.5 ft

17. d = 221.25 m

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Archimedes

Archimedes was a Greek mathematician and philosopher. Among many other things, he identified 3.14 as pi.Circumference

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

Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.Pi

(Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.### Image Attributions

Here you'll learn to identify pi as the ratio of diameter to circumference in a circle.