10.7: Volume of Cylinders
Introduction
Sports Bottle Dilemma
“I need a new sports bottle and I have two picked out, but I am trying to figure out which one is better,” Trevor told Candice one morning between wrapping jobs.
“What makes one better than the other?” Candice asked.
“Well, I want the bottle with the greatest volume,” Trevor said.
“Can’t you figure that out by the number of ounces on the package?” Candice asked.
“Well, I could, but I want to know the cubic inches,” Trevor said. “The one with more cubic inches would be the better choice. I want to use math to figure it out.”
“Alright, well you need the height and the radius of each bottle to do it that way. It definitely seems like more work to me, but I can get using math to test it all out.”
“Here are the dimensions that I have,” Trevor said taking out a piece of paper. He handed it to Candice.
On the paper Trevor has written the following dimensions.
Bottle #1 – Height = 7.5" Diameter = 5"
Bottle #2 – Height = 9" Diameter = 3"
“First, you need the radius, so divide each diameter in half,” Candice said looking at the paper.
While Trevor works on this, you will need to take some time to learn to use formulas to find the volume of cylinders. Then at the end of the lesson you will be able to help Candice and Trevor with the sports bottle dilemma.
What You Will Learn
By the end of this lesson, you will be able to apply the following skills:
- Recognize volume of cylinders as the sum of volumes of layers of unit cubes.
- Find volumes of cylinders using formulas
- Find heights of cylinders given volumes and another dimension
- Solve real-world problems involving volumes of cylinders.
Teaching Time
I. Recognize Volume of Cylinders as the Sum of Volumes of Layers of Unit Cubes
In this lesson we will learn to find the volume of cylinders. A cylinder is a solid shape that exists in three-dimensional space. A cylinder has two faces that are circles. We do not call the side of a cylinder a face because it is curved. We still have to include its area in the total surface area of the cylinder, however. We see cylinders in everyday life all the time. Here is the basic shape of a cylinder, think about how many other places you can see them in the world around you.
The volume of a cylinder is the measure of how much three-dimensional space it takes up or holds. Imagine a thermos. Its size determines how much water the thermos will hold. If we fill it with water, the amount of water tells the volume of the thermos. We measure volume in cubic units, because we are multiplying three dimensions: length, width, and height. The width of the thermos is the same as the diameter of the circular faces.
There are different ways to examine and think about volume. One way is to use unit cubes. Look at this diagram of a cylinder to understand this concept.
Here you can see that the unit cubes are being used to fill up the cylinder. Remember that because we are talking about volume, we are measuring the space contained inside the cylinder. This can be tricky with a cylinder because the bases are round-squares aren’t exactly round, so we should find another way to measure the volume of a cylinder.
II. Find Volumes of Cylinders using Formulas
Let’s think about what information we are going to need to gather to figure out the volume of a cylinder. We know that the two bases are circular, so we are going to need to know the area of the circle to figure out how much space can be contained on top of it. A cylinder is tall or has height, so we are going to need to know the height of the cylinder. Then we will know how high the space is inside the cylinder.
First, let’s think about the area of the circular bases. To find the area of a circle, we can use the following formula.
\begin{align*}A= \pi r^2 \end{align*}
But we also have the height \begin{align*}(h)\end{align*}
\begin{align*}V= \pi r^2 h \end{align*}
Take a minute and copy this formula down in your notebook.
Now let’s look at applying this formula as we work with a few examples.
Example
Find the volume of a cylinder with a radius of 5 cm and a height of 7 cm.
We can start by substituting the values of the cylinder into our formula.
\begin{align*}V & = \pi r^2 h \\
V & = (3.14)(5^2 )(7) \\
V & =(3.14)(25)(7) \\
V & = 549.5 \ cm^3 \end{align*}
This is our answer. Notice that we measured the volume of the cylinder in cubic units because we are multiplying three different measurements.
Let’s look at another one.
Example
Find the volume of a cylinder with a diameter of 12 inches and a height of 8 inches.
First, notice that we have been given the diameter and not the radius. We can divide twelve inches in half and that will give us the radius of the circular base of the cylinder.
Radius = 6 inches.
Now we can substitute the values into the formula and solve for the volume.
\begin{align*}V & = \pi r^2 h \\
V & = (3.14)(36)(8) \\
V & = 904.32 \ in^3 \end{align*}
This is our answer.
10K. Lesson Exercises
Find the volume of each cylinder.
- radius = 3 in, height = 7 in
- radius = 2.5 mm, height = 4 mm
- diameter = 14 in, height = 9 in
Check your answers with a neighbor. Be sure that your answers are correct before moving on to the next section.
III. Find Heights of Cylinders Given Volumes and another Dimension
Sometimes, you will know the volume of a cylinder and you won’t know the height of it. Think about a water tower that is cylindrical in shape. We might know how much volume the tank will hold, but not the height of it. When this happens, we can use our formula to find the missing height of the cylinder.
Example
A cylinder with a radius of 2 inches has a volume of 125.6 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 volume and the radius. We put these into the formula and then solve for \begin{align*}h\end{align*}, the height.
\begin{align*}V & = \pi r^2h\\ 125.6 & = \pi (2)^2 h\\ 125.6 & = 4 \pi h\\ 125.6 & = 12.56 \ h\\ 125.6 \div 12.56 & = h\\ 10 \ in. & = h\end{align*}
The height of the cylinder is 10 inches.
We used the volume formula to solve for \begin{align*}h\end{align*} and found that the height of the cylinder is 10 inches. We can check our work by putting this number in for the height. We should get a volume of 125.6 cubic inches.
\begin{align*}V & = \pi r^2h\\ V & = \pi (2)^2 (10)\\ V & = \pi (4) (10)\\ V & = 40 \pi\\ V & = 125.6 \ in.^3\end{align*}
Our calculation was correct! Let’s look at another example.
Example
What is the height of a cylinder that has a radius of 6 cm and a volume of \begin{align*}904.32 \ cm^3\end{align*}?
Again, we have been given the volume and the radius. We put this information into the formula and solve for \begin{align*}h\end{align*}, the height.
\begin{align*}V & = \pi r^2h\\ 904.32 & = \pi (6)^2 h\\ 904.32 & = 36 \pi h\\ 904.32 & = 113.04 \ h\\ 904.32 \div 113.04 & = h\\ 8 \ cm & = h\end{align*}
The height of this cylinder is 8 centimeters.
10L. Lesson Exercises
Find the height of each cylinder given the radius and volume.
- \begin{align*}r = 6 \ in, \ \text{Volume} = 904.32 \ in^3\end{align*}
- \begin{align*}r = 3 \ m, \ \text{Volume} = 254.34 \ m^3\end{align*}
- \begin{align*}r = 5 \ ft, \ \text{Volume} = 785 \ ft^3\end{align*}
Go over your answers with a peer and then move on to the next section.
IV. Solve Real-World Problems Involving Volumes of Cylinders
We can also use what we have learned to find the volume of cylinders that we use in real-life. Be sure you understand what the problem is asking. Then look to see what information is given in the problem. Third, put this information in for the appropriate variable in the formula and solve. Let’s give it a try.
Example
A water tank has a radius of 50 feet and a height of 400 feet. How many cubic feet of water will the tank hold when it is full?
First, let’s determine what the problem is asking us to find. We need to find the volume of the tank, which is the amount of water it can hold. What information have we been given? We know the radius and the height of the tank, so we can put this information into the formula and solve for \begin{align*}V\end{align*}, volume.
\begin{align*}V & = \pi r^2h\\ V & = \pi (50)^2 (400)\\ V & = \pi (2,500) (400)\\ V & = 1,000,000 \pi\\ V & = 3,140,000 \ ft^3\end{align*}
The water tank will hold more than 3 million cubic feet of water!
Example
Javier wants to construct a cylindrical container to hold enough water for his pet fish. He read that the fish needs to live in 2,110.08 cubic inches of water. If he constructs a tank with a diameter of 16 inches, how tall must he make it so that it holds the right amount of water?
Again, the first thing we need to do is decide what the problem is asking us to find. We need to know how tall Javier must make the tank. In other words, we need to find the height of the cylinder.
What information have we been given? We know that the tank must hold 2,110.08 cubic inches of water. This is the volume. What about the radius? We know that the diameter of the tank will be 16 inches, so the radius will be \begin{align*}16 \div 2 = 8\end{align*} inches. We put the radius and volume into the volume formula and solve for \begin{align*}h\end{align*}, the height.
\begin{align*}V & = \pi r^2h\\ 2,110.08 & = \pi (8)^2 h\\ 2,110.08 & = 64 \pi h\\ 2,110.08 & = 200.96h\\ 2,110.08 \div 200.96 & = h\\ 10.5 \ in. & = h\end{align*}
In order for his tank to hold 2,110.08 cubic inches of water, Javier must make his tank 10.5 inches tall.
Great job! We can solve all kinds of problems using the volume formula. Just be sure you know what the problem is asking you to find and what information has been given.
Real–Life Example Completed
Sports Bottle Dilemma
Here is the original problem once again. Think about what you have learned as Trevor and Candice figure out the volume of each sports bottle.
“I need a new sports bottle and I have two picked out, but I am trying to figure out which one is better,” Trevor told Candice one morning between wrapping jobs.
“What makes one better than the other?” Candice asked.
“Well, I want the bottle with the greatest volume,” Trevor said.
“Can’t you figure that out by the number of ounces on the package?”Candice asked.
“Well, I could, but I want to know the cubic inches,” Trevor said. “The one with more cubic inches would be the better choice. I want to use math to figure it out.”
“Alright, well you need the height and the radius of each bottle to do it that way. It definitely seems like more work to me, but I can get using math to test it all out.”
“Here are the dimensions that I have,” Trevor said taking out a piece of paper. He handed it to Candice.
On the paper Trevor has written the following dimensions.
Bottle #1 – Height = 7.5" Diameter = 5"
Bottle #2 – Height = 9" Diameter = 3"
“First, you need the radius, so divide each diameter in half,” Candice said looking at the paper.
After dividing each diameter, here are the new dimensions.
Bottle #1 = H = 7.5" Radius = 2.5"
Bottle #2 = H = 9" Radius = 1.5"
Next, we use the formula to find the volume of bottle #1.
\begin{align*}V & = \pi r^2 h \\ V & = 3.14(2.5^2)(7.5) \\ V & = 147.18 \ cubic \ inches \end{align*}
Next, we find the volume of bottle #2.
\begin{align*}V & = \pi r^2 h \\ V & = 3.14(1.5^2)(9) \\ V & = 63.58 \ cubic \ inches \end{align*}
Wow! Notice the difference in volumes! Even though the second bottle was taller, the diameter was smaller and this greatly impacted the volume of the bottle. The first bottle holds more than twice as much as the second bottle!!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Volume
- the amount of space contained inside a three-dimensional solid. Volume is sometimes used to measure fluid capacity.
- Cubic Units
- how we measure volume. It is measured in cubic units because we multiply the length, the width and the height of a solid.
- Cylinder
- a solid with two circular bases and a curved side.
Technology Integration
Khan Academy Cylinder Volume and Surface Area
- http://www.mathplayground.com/mv_volume_cylinders.html – This is a Brighstorm video on how to find the volume of a cylinder.
Time to Practice
Directions: Given the radius and height of each cylinder, find its volume.
1. \begin{align*}r = 5 \ in, \ h = 8 \ in\end{align*}
2. \begin{align*}r = 4 \ in, \ h = 7 \ in\end{align*}
3. \begin{align*}r = 3 \ ft, \ h = 5 \ ft\end{align*}
4. \begin{align*}r = 3 \ ft, \ h = 8 \ ft\end{align*}
5. \begin{align*}r = 4 \ cm, \ h = 9 \ cm\end{align*}
6. \begin{align*}r = 6 \ m, \ h = 12 \ m\end{align*}
7. \begin{align*}r = 7 \ in, \ h = 14 \ in\end{align*}
8. \begin{align*}r = 5 \ m, \ h = 10 \ m\end{align*}
9. \begin{align*}r = 2 \ m, \ h = 11 \ m\end{align*}
10. \begin{align*}r = 3 \ cm, \ h = 12 \ cm\end{align*}
Directions: Given the volume and the radius, find the height of each cylinder.
11. \begin{align*}r = 6 \ in, \ V = 904.32 \ in^3\end{align*}
12. \begin{align*}r = 5 \ in, \ V = 706.5 \ in^3\end{align*}
13. \begin{align*}r = 7 \ ft, \ V = 2307.9 \ ft^3\end{align*}
14. \begin{align*}r = 8 \ ft, \ V = 4019.2 \ ft^3\end{align*}
15. \begin{align*}r = 7 \ ft, \ V = 1538.6 \ ft^3\end{align*}
16. \begin{align*}r = 12 \ m, \ V = 6330.24 \ m^3\end{align*}
17. \begin{align*}r = 9 \ m, \ V = 4069.49 \ m^3\end{align*}
18. \begin{align*}r = 10 \ m, \ V = 5652 \ m^3\end{align*}
19. \begin{align*}r = 12 \ in, \ V = 11304 \ in^3\end{align*}
Directions: Solve each word problem.
20. A cylinder with a radius of 7 meters has a volume of 769.3 cubic meters. What is its height?
21. The cylindrical pool in Wayne’s backyard holds 15,700 cubic feet of water. If the diameter of the pool is 50 feet, how deep is the pool?
22. What is the volume of Keisha’s thermos if it has a radius of 2.5 in at the opening and 10 in for a height?
23. Mr. Riley bought 2 cans of paint to paint his garage. Each can had a radius of 5.5 inches and a height of 8 inches. How many cubic inches of paint did he buy in all?
24. For a science project, Jason put a can out to collect rainwater. The can was 11 inches tall and had a diameter of 8 inches. If the can caught 125.6 cubic inches each day, how many days did it take to fill the can?
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