6.2 Circumference & Area of Circles

Learning Objectives

You will demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle.
You will show you understanding of $\pi$ .
You will calculate the circumference and area of circles.
You will be able to find the sectors of circles and solve problems in various contexts.

Introduction

Pitchers on Deck

Miguel’s 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. Pitchers practice their warm-ups while 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.

While Miguel does this, it is time for you to learn about circumference and area. By the end of this lesson you will know how to find the circumference and area of each of the circles just like Miguel does.

Guided Practice

First we will learn how to find the circumference of a circle given the radius or diameter

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

In lesson 3.1 you learned 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.

Example A

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.

$C & = \pi d\\C & = \pi (8)\\C & = 25.12 \ in.$

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.

Example B

What is the circumference of the circle below?

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

$C & = \pi d\\C & = \pi (12.7)\\C & = 39.88 \ m$

We can 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 two and end up with the same measure as the diameter. Here is how we can alter the formula when given a radius.

$C=2\pi r$

Now let’s try it out with an example.

Example C

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.

Try a few on your own.

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

$d = 5 \ in$
$r = 3.5 \ in$
$d = 10 \ ft$ .

Check your answers with a neighbor.

Now write down the formulas for finding the circumference of a circle given the radius or the diameter.

Example D

Now we will learn how to find the diameter or radius given the circumference.

Sometimes a problem will give us the circumference of a circle and ask us to find either its diameter or its radius. We can still use the formula for circumference. All we have to do is put the information we have into the appropriate place in the formula and solve for the unknown quantity.

A circle has a circumference of 20.72 m. What is its diameter?

In this problem, we are given the circumference and we need to find the diameter. We put these numbers into the formula and solve for .

$C & = \pi d\\20.72 & = \pi d\\20.72 \div \pi &= d\\6.6 & = d$

By solving for , we have found that the diameter of the circle is 6.6 meters.

Let’s check our calculation to be sure. We can check by putting the diameter into the formula and solving for the circumference:

$C & = \pi d\\C & = \pi (6.6)\\C & = 20.72 \ m$

We know the circumference is 20.72 meters, so our calculation is correct.

Example E

The circumference of a circle is 147.58 yards. Find its radius.

Again, we have been given the circumference. Read carefully! This time we need to find the radius, not the diameter. We can use the formula for radii and solve for .

$C & = 2 \pi r\\147.58 & = 2 \pi r\\147.58 & = 6.28r\\147.58 \div 6.28 & = r\\23.5 \ yd & = r$

We have found that the circle has a radius of 23.5 yards.

This time let’s try checking our work by using the other formula to find the diameter. Remember the diameter is twice the length of the radius.

$C & = \pi d\\147.58 & = \pi d\\147.58 \div \pi & = d\\47 \ yd & = d$

We have found that the diameter of the circle is 47 yards. The radius must be half this length, or $47 \div 2 = 23.5$ yards.

Our calculation is correct!

Whenever we are given the circumference, we can use the formula to solve for the diameter or the radius. The number for pi always stays the same, so we only need one piece of information about a circle to find the other measurement.

Now it’s your turn.

Find the missing dimension.

The circumference is 28.26 inches. What is the diameter?
The circumference is 21.98 feet. What is the radius?
The circumference is 34.54 meters. What is the diameter?

Take a few minutes to go over your answers with a peer.

Example F

Recognize the formula for the area of a circle

To find the area of a two-dimensional figure, we need to figure out the measurement of the space contained inside.This is the measurement of area. This is also the measurement inside a circle. You learned how to find the radius of a circle, given circumference or diameter. now let’s look at using the radius to find the area of the circle.

How do we find the area of a circle?

The area of a circle is found by taking the measurement of the radius, squaring it and multiplying it by pi.

Here is the formula.

$A= \pi r^2$

Write this formula in your notebook.

Find the Area of Circles Given the Radius or Diameter

We already know that the symbol $\pi$ represents the number 3.14, so all we need to know to find the area of a circle is its radius. We simply put this number into the formula in place of and solve for the area, .

What is the area of the circle below?

We know that the radius of the circle is 12 centimeters. We put this number into the formula and solve for .

$A & = \pi r^2\\A & = \pi (12)^2\\A & = 144 \ \pi\\A & = 452.16 \ cm^2$

Remember that squaring a number is the same as multiplying it by itself.

The area of a circle with a radius of 12 centimeters is 452.16 square centimeters.

Try another problem applying the formula for area of a circle

Some students have formed a circle to play dodge ball. The radius of the circle is 21 feet. What is the area of their dodgeball circle?

The dodgeball court forms a circle, so we can use the formula to find its area. We know that the radius of the circle is 21 feet, so let’s put this into the formula and solve for area, .

$A & = \pi r^2\\A & = \pi (21)^2\\A & = 441 \ \pi\\A & = 1,384.74 \ ft^2$

Notice that a circle with a larger radius of 21 feet has a much larger area than the circle with a 12 centimeter radius: 1,384.74 square feet.

Sometimes, you will be given a problem with the diameter and not the radius. When this happens, you can divide the measurement of the diameter by two and then use the formula.

Here is another example of area but this time the diameter is given instead of the radius.

Find the area of a circle with a diameter of 10 in.

First, we divide the measurement in half to find the radius.

$10 \div 2 = 5 \ in$

Now we use the formula.

$A & = \pi r^2 \\A& = 3.14(5^2) \\A&=3.14 (25) \\A&=78.5 \ square \ inches$

Try a few on your own.

Find the area of each circle.

Radius = 9 inches
Radius = 11 inches
Diameter = 8 ft.

Take a few minutes to check your work.

Example G

Find the radius or diameter given the area

We have seen that when we are given the radius or the diameter of a circle, we can find its area. We can also use the formula to find the radius or diameter if we know the area. Let’s see how this works.

The area of a circle is 113.04 square inches. What is its radius?

This time we know the area and we need to find the radius. We can put the number for area into the formula and use it to solve for the radius, .

$A & = \pi r^2\\113.04 & = \pi r^2\\113.04 \div \pi & = r^2\\36 & = r^2\\\sqrt{36} & = r \\6 \ in. & = r$

Let’s look at what we did to solve this. To solve this problem we needed to isolate the variable . First, we divided both sides by $\pi$ , or 3.14. Then, to remove the exponent, we took the square root of both sides. A square root is a number that, when multiplied by itself, gives the number shown. We know that 6 is the square root of 36 because $6 \times 6 = 36$ .

The radius of a circle with an area of 113.04 square inches is 6 inches.

Let's try another problem where we have to work backwards using the formula for area!

What is the diameter of a circle whose area is $379.94 \ cm^2$ ?

What is this problem asking us to find? We need to find the diameter (not the radius!). What information is given in the problem? We know the area. Therefore we can use the formula to solve for the radius, . Once we know the radius, we can find the diameter.

$A & = \pi r^2\\379.94 & = \pi r^2\\379.94 \div \pi & = r^2\\121 & = r^2\\\sqrt{121} & = r\\11 \ cm & = r$

The radius of a circle with an area of 379.94 square centimeters is 11 centimeters.

Remember, this problem asked us to find the diameter, so we’re not done yet. How can we find the diameter?

The diameter is always twice the length of the radius, so the diameter of this circle is $11 \times 2 = 22$ centimeters.

As we have seen, we can use the area formula with lots of kinds of information about a circle. If we know the diameter or radius, we can solve for the area, . If we are given the area, we can solve for the radius, . If we know the radius, we can also find the diameter.

Try a few more.

Find the radius of each circle.

Area = 153.86 sq. in.

Area = 354.34 sq. ft.

452.16 sq. m

Check your work with a partner and then continue with the next section.

Example H

Solve real world problems involving the circumference of circles

We have seen that we can apply the formula for finding the circumference of circles to different kinds of situations. Sometimes we need to solve for circumference, but other times we may need to find the diameter or the radius. We can also use this formula when we are given real measurements. Let’s try a few problems involving circles in the real world.

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

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 . 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.

$C & = \pi d\\C & = \pi (9)\\C & = 28.26 \ in.$

The circumference of the pie is 28.26 inches.

Good job! Let’s move on.

Florencia wants to paste some ribbon around a circular mirror to make a border. The mirror is 40 inches across. If the ribbon is sold by the inch and costs $0.15 per inch, how much will Florencia need to spend to buy enough ribbon?

What is this problem asking us to find? We need to find how much money Florencia will spend on the ribbon.

In order to determine this, we first need to know how much ribbon is necessary to go around the mirror. Therefore we need to find the circumference of the mirror. We know that it is 40 inches across. Is this the radius or the diameter? It is the diameter, so we can put this information into the formula and solve for the circumference.

$C & = \pi d\\C & = \pi (40)\\C & = 125.6 \ in.$

The circumference of the mirror is 125.6 inches, so Florencia will need 125.6 inches of ribbon to put around it.

We’re not done yet, however. Remember, we need to find how much money she will spend to buy the ribbon.

Because the ribbon is sold by the inch, Florencia will need to buy 126 inches. We know that it is sold at $0.15 an inch, so we simply multiply the number of inches by the cost per inch.

$126 \ inches \times \$ 0.15 \ per \ inch = \$ 18.90$

Florencia will need to spend $18.90 in order to buy enough ribbon to go around her mirror.

Real Life Example Completed

Pitchers on Deck

Here is the original problem once again. Reread it and underline any important information.

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. Pitchers practice their warm-ups while standing on them. They work on stretching and get ready to “pitch” the ball prior to their turn on the mound.