“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 one that holds the most 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 about volume and cylinders. In this Concept, you will learn to use formulas to find the volume of cylinders. Then at the end of the Concept, you will be able to help Candice and Trevor with the sports bottle dilemma.**

### Guidance

In this Concept 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

**, 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.**

*cubic units*
**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.

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*} of the cylinder to figure into this calculation. If we put all of these parts together, we have the following formula.**

\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 in a few situations.

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.

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.**

Find the volume of each cylinder.

#### Example A

radius = 3 in, height = 7 in

**Solution: \begin{align*}197.82 \ in^3\end{align*}**

#### Example B

radius = 2.5 mm, height = 4 mm

**Solution: \begin{align*}78.5 \ mm^3\end{align*}**

#### Example C

diameter = 14 in, height = 9 in

**Solution: \begin{align*}1384.74 in^3\end{align*}**

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 one that holds the most 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

- Volume
- the amount of space contained inside a three-dimensional solid. Volume is often used to measure capacity or liquid.

- 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 flat rounded side.

### Guided Practice

Here is one for you to try on your own.

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?

**Answer**

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 \ in.^3\end{align*}

**The water tank will hold more than 3** *million***cubic feet of water!**

### Video Review

- This is a Khan Academy video on solid geometry volume.

### 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*}

11. \begin{align*}r = 6 \ cm, \ h = 11 \ cm\end{align*}

12. \begin{align*}r = 4 \ m, \ h = 14 \ m\end{align*}

13. \begin{align*}r = 13 \ cm, \ h = 26 \ cm\end{align*}

14. \begin{align*}r = 8 \ in, \ h = 14 \ in\end{align*}

15. \begin{align*}r = 4.5 \ cm, \ h = 16.5 \ cm\end{align*}