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# LESSON: Circumference of Circles

Circumference of Circles

 Site: Mountain Heights Academy OpenCourseWare Course: Mathematics Essentials Q4 v2013 Book: LESSON: Circumference of Circles Printed by: Guest user Date: Friday, 12 May 2017, 5:17 PM

## Objectives

• Identify ratio of circumference to diameter as pi.
• Find the circumference of circles given diameter or radius.
• Find diameter or radius of circles given circumference.
• Solve real world problems involving circumference of circles.

## Vocabulary

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.
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.

## Identify Ratio of Circumference to Diameter as Pi

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 distance around the outer edge of the circle too. This distance is called the circumference of the circle. The circumference of the circle is the perimeter of the circle, only with circles we don’t call it perimeter we call the measure around the outside edge of the circle the circumference.

To understand things about circles, let’s consider some history.

Back in the 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 distance across the circle or the diameter of the circle and the distance around the edge of the circle or the circumference of the 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, well the circumference of the circle is three and a little more times the 5 cm.

Because of that, we say that the ratio of the diameter to the circumference is pi. We use the number 3.14 for pi because the ratio is a terminating decimal and does not end. However we use two decimal places for pi works for figuring out the circumference of the circle.

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

1. Who was the first person to figure out the relationship between the diameter and the circumference?
2. What is the distance across the circle called?
3. What is the distance around the circle called?

## Find the Circumference of a Circle Given the Diameter or Radius

Now that you know about the relationship between the diameter of a circle and the circumference of the circle, we can work on figuring out the circumference using a formula and pi.

To figure out the circumference of the circle, we multiply the diameter of the circle times pi or 3.14.

$C = d\pi$

Remember, whenever you see the symbol for pi, you substitute 3.14 in when multiplying.

Example 1

Find the circumference.

The diameter of the circle is 6 inches. We can substitute this given information into our formula and solve for the circumference of the circle.

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

What about if we have been given the radius and not the diameter?

Example 2

Find the circumference.

Remember that the radius is one-half of the diameter. You can solve this problem in two ways.

1. Double the radius right away and then use the formula for diameter to find the circumference.
2. Use this formula when you have been given the radius of the circle.

$C = 2\pi r$

Let’s use the formula to find the circumference of the circle.

$C & = 2(3.14)(4)\\ C & = 3.14(8)\\ C & = 25.12 \ cm$

## Find the Diameter or Radius of a Circle Given Circumference

What happens if you are given the circumference but not the radius or the diameter? Can you still solve for one or the other?

Working in this way is a bit tricky and will require us to work as detectives once again. You will have to work backwards to figure out the radius and/or the diameter when given only the circumference to work with.

Example 1

Find the diameter of a circle with a circumference of 21.98 feet.

To work on this problem, we will need our formula for finding the circumference of a circle.

$C = \pi d$

Next, we fill in the given information.

$21.98 = (3.14)d$

To solve this problem we need to figure out what times 3.14 will give us 21.98. To do this, we divide 21.98 by 3.14.

${3.14 \overline{ ) {21.98 \;}}}$

Remember dividing decimals? First, we move the decimal point two places to make our divisor a whole number. Then we can divide as usual.

$& \overset{ \qquad \ \quad 7}{314 \overline{ ) {2198}}}$

The diameter of this circle is 7 feet.

How could we figure out the radius once we know the diameter?

We can figure out the radius by dividing the diameter in half. The radius is one-half the measure of the diameter.

7 $\div$ 2 $=$ 3.5

The radius of the circle is 3.5 feet.