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# Volume of Cylinders

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Have you ever wondered how much soup is in one can? Take a look at this dilemma.

In the midst of all of the painting, Jose and Carmen sent Alicia out to the grocery store to pick up some lunch. Since the three were painting at Jose’s house, his mother had made some bread and soup seemed like an obvious choice.

When Alicia got to the store, she wasn’t sure how much soup to buy. She picked out a yummy looking organic chicken vegetable soup that she was sure everyone would like, but she couldn’t decide between two and three cans.

After standing there for a few minutes, her stomach began grumbling and she decided to go with the three cans.

“If there is any extra, someone at Jose’s will eat it,” she thought to herself.

Alicia bought the three cans of soup and headed back to Jose’s house.

Each can had a diameter of 5.4 inches and a height of 6.7 inches. What is the total volume of soup that Alicia bought?

To figure this problem out, you will need to know about volume. Pay close attention to the work being done in this Concept and by the end of it, you will know how to answer this question.

### Guidance

What is volume?

Volume of a solid figure is the measure of how much three-dimensional space it takes up or holds.

Imagine an aquarium for fish. Its length, width, and height determine how much water the tank will hold. If we fill it with water, the amount of water tells the volume of the tank.

We measure volume in cubic units, because we are multiplying three dimensions: length, width, and height.

There are several different ways that we can calculate volume. The first way that we are going to explore is by looking at we can calculate volume using unit cubes.

What are unit cubes?

Unit cubes are singular cubes used to represent one unit. When we “fill up” a solid figure with unit cubes, we can see the unit cubes lining up the figure. Then we can count or calculate the number of unit cubes in the solid. The number of unit cubes in the solid figure is the volume of the figure.

We can use unit cubes to calculate the volume of a cylinder.

This one will be a little funnier because a cylinder is circular and unit cubes are squares. This means that since we can’t cut a unit cube into sections, that it will be impossible for us to calculate an accurate measurement for volume. We will be estimating the volume of the cylinder.

You can see that it doesn’t always work to calculate the volume of a figure by counting unit cubes. You can think about this in the case of the cylinder. There is an easier way. We can calculate the volume of any solid figure by using a formula.

Now let's look at the formula.

Write this formula down in your notebook.

All we need to know is the radius of the circular faces and the height of the cylinder. We simply put these numbers into the formula and solve for volume, $V$ . Let’s give it a try.

Find the volume of the cylinder below.

We have been given all the information we need in order to solve for volume. Let’s put the numbers into the formula.

$V &= \pi r^2 h\\V &=\pi(5.5^2)(2.7)\\V &=\pi(30.25)(2.7)\\ V &=81.68 \pi\\V &=256.48 \ cm^3$

The volume of this cylinder is 256.48 cubic centimeters.

We can also think about working backwards if we have the volume and one other dimension. Then we can problem solve to figure out the missing dimension.

A cylinder with a radius of 3 inches has a volume of $140.4 \pi$ cubic inches. What is the height of the cylinder?

What is the problem asking us to find? We need to solve for the height of the cylinder. The problem tells us the radius and the volume. This time the volume is written as a function of pi. This is a way of showing a more specific number, rather than approximating with 3.14. We simply put the whole number into the formula for $V$ and then solve for $h$ , the height.

$V &= \pi r^2h\\140.4 \pi &= \pi (3^2) h\\140.4 \pi &= 9 \pi h\\140.4 \pi \div 9 \pi &= h \quad \text{Divide both sides by} \ 9 \pi. \ \pi \ \text{cancels out.}\\15.6 \ in. &= h$

We used the volume formula to solve for $h$ and found that the height of the cylinder is 15.6 inches.

Find the volume for each cylinder.

#### Example A

A cylinder with a radius of 3 inches and a height of 9 inches.

Solution: $254.34$ sq.inches

#### Example B

A cylinder with a radius of 2.5 feet and a height of 6 feet.

Solution: $117.75$ sq. feet

#### Example C

A cylinder with a diameter of 5 inches and a height of 7 inches.

Solution: $137.37$ sq. inches

Now let's go back to the dilemma from the beginning of the Concept.

What do we need to find?

We need to find the volume of soup that Alicia bought. Keep in mind that she bought 3 cans of soup. We need to find the volume of one can of soup and then multiply this amount by 3 to find the total volume.

What information have we been given?

First, we know that the soup cans are cylinders, so we’ll need to use the volume formula for cylinders. We also know that the height of each can is 6.7 inches. What is the radius? We have only been given the diameter, which is 5.4 inches. Therefore we need to divide by 2 to find the radius.

$5.4 \div 2 = 2.7$

The radius of each can is 2.7 inches. Now we can put this information into the formula and solve for $V$ , volume.

$V &= \pi r^2h\\V &= \pi (2.7^2)(6.7)\\V &=\pi (7.29)(6.7)\\V &= 48.84 \pi\\ V &= 153.36 \ in.^3$

Each can has a volume of 153.36 cubic inches when we approximate pi as 3.14.

But we’re not done yet! Remember, we need to find the total volume of three cans of soup. Therefore we need to multiply the volume of one can by 3.

$153.36 \times 3 = 460.08 \ in^3$

Alicia bought a total of 460.08 cubic inches of soup.

### Vocabulary

Volume
the amount of water or capacity that a solid figure can hold. Volume is measured in cubic units.

### Guided Practice

Here is one for you to try on your own.

Find the volume of the following cylinder.

A water tank has a radius of 45 feet and a height of 300 feet. How many cubic feet of water will the tank hold when it is full?

Solution

To solve this problem, we use the formula for finding the volume of a cylinder and substitute the given values into that formula.

$V &= \pi r^2h\\V &= \pi (45^2)(300)\\V &=\pi (2025)(300)\\V &= 607,500 \pi\\ V &= 1,907,550 \ ft.^3$

This is the answer.

### Explore More

1. Name the figure pictured above.
2. What is the diameter of the figure?
3. What is the radius is the figure?
4. What is the volume of the figure?

Directions: Find the volume of each cylinder given the radius and height.

1. $r = 4 feet, h = 5 feet$
2. $r = 6 cm, h = 8 cm$
3. $r = 4.5 feet, h = 5 feet$
4. $r = 3.5 m, h = 7 m$
5. $r = 13 ft, h = 2 ft$
6. $r = 11 m, h = 12 m$
7. $r = 1.5 ft, h = 3 ft$
8. $r = 7 in, h = 12 in$
9. $r = 8 cm, h = 11 cm$
10. $r = 5 m, h = 9 m$
11. $r = 4.5 ft, h = 6 ft$