6.1 Relationship Between Diameter And Circumference

Learning Objectives

You will identify the diameter and the radius on a circle.
You will calculate how to find the distance around the circle by using other measurements within the circle.
You will be familiar with using the calculator to determine circumference.
You will have a clear understanding of the value of Pi.

Introduction

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? 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.

Guided Learning

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.

Example

We can see the ratio 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.

$\frac{\text{circumference}}{\text{diameter}} \quad = \quad \frac{6.28}{2} \quad = \quad 3.14 && \frac{\text{circumference}}{\text{diameter}} \quad = \quad \frac{12.56}{4} \quad = \quad 3.14$

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 $\pi$ . 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.

$\frac{C}{d}= \pi$

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

$C = \pi d$

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

Find the circumference of the following circle.

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

$C&=2 \pi (3) \\C&=2(3.14)(3) \\C&=3.14(6) \\C&=18.84 \ inches$

This is our answer.

Real Life Problem Completed

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

$C &=\pi(8)\\C &=3.14(8)\\C &=25.12 \ feet$

Review

C, d, and r are all symbols used to represent calculations of circles or parts of a circle.

Circumference

The distance around a circle is called the circumference.

Perimeter

Perimeter is the distance around a two-dimensional shape.

Circle

A circle is a closed plane curve consisting of all points at a given distance from a point within it called the center.

Radius

The distance from the center of the circle to the outside edge of the circle is called the radius.

Diameter

The distance across the center of the circle is called the diameter.

Pi is a name given to the ratio of the circumference of the circle to the diameter.

Video Resources

Khan Academy Circumference of a Circle

http://www.mathplayground.com/mv_circumference.html