# 10.18: Volume of Cylinders

**At Grade**Created by: CK-12

**Practice**Volume of Cylinders

Have you ever compared jars and wondered which one held more? Take a look at this dilemma.

Jillian’s grandmother loves to cook. One day in between sewing projects, she takes Jillian to the grocery store and comes home with a big bag of assorted beans. Grandma’s baked beans are Jillian’s favorite, and she is thrilled that Grandma is going to cook them for dinner.

Jillian takes two different jars from the cupboard. One is long and thin and one is wide.

“Which jar should I use?” Jillian asks her grandmother.

“Use whichever one will hold the most,” her grandmother says.

Jillian looks at the two jars. Here is what they look like.

Jillian has measured each jar to try to figure out which one will hold the most. She just isn’t sure what to do now.

**This is your task. Jillian will need to figure out the volume of each cylinder. This Concept will teach you all about calculating volume. Calculate the volume of each cylinder and then you will know which one will hold the most beans.**

### Guidance

**Volume is the amount of space contained within a solid figure.** Since cylinders often contain liquid, you can imagine that the volume of cylinders often has to do with some kind of liquid. In the case of cylinders, you can think of volume as capacity.

Here is a cylinder that is probably used in a science lab. Here volume would be compared with capacity of liquid.

Here is a picture of a boy swimming in a pool. When you think about volume in this case, it is the capacity of the pool. The volume would be the amount of water in the pool.

This paint can is a cylinder. If we wanted to figure out the volume of this cylinder, we would need to figure out the amount of space inside the paint can. This would be the volume of the cylinder.

We can think about the volume of a cylinder as we would think about the volume of a prism. We can use unit cubes to fill a cylinder.

You can see that we have started to fill this cylinder with cubes to calculate the volume. The problem is that the cubes don’t fit perfectly inside the cylinder. To calculate the volume of a cylinder accurately, we need to use a formula.

**Which formula can we use to calculate the volume of a cylinder?**

To calculate the volume of a cylinder, we need to calculate the area of the circular base. That will give us a measure for the number of unit cubes that can fit across the bottom of the cylinder. The height of the cylinder will show us how high cubes can be stacked inside the cylinder.

**Here is the formula for finding the volume of a cylinder.**

\begin{align*}V = \pi r^2h\end{align*}

Let's apply this formula.

The radius of the circular base is 2 inches. The height of the cylinder is 7 inches. If we take both of these given measures and substitute them into the formula, we can solve for the volume of the cylinder.

\begin{align*}V & = \pi r^2h\\ V & = (3.14)(2^2)(7)\\ V & = (3.14)(4)(7)\\ V & = (3.14)(28)\\ V & = 87.92 \ in^3\end{align*}

**The volume of the cylinder is \begin{align*}87.92 \ in^3\end{align*}.**

Use the formula to find the volume of the following cylinders.

#### Example A

**Solution: \begin{align*}25.12 in^3\end{align*}**

#### Example B

\begin{align*}d = 10 \ ft, \ h = 12 \ ft\end{align*}

**Solution: \begin{align*}942 ft^3\end{align*}**

#### Example C

\begin{align*}r = 6 \ in, \ h = 10 \ in\end{align*}

**Solution:\begin{align*}1130.4 in^3\end{align*}**

Here is the original problem once again. Use what you have learned about the volume of cylinders to help Jillian problem solve this dilemma.

Jillian’s grandmother loves to cook. One day in between sewing projects, she takes Jillian to the grocery store and comes home with a big bag of assorted beans. Grandma’s baked beans are Jillian’s favorite, and she is thrilled that Grandma is going to cook them for dinner.

Jillian takes two different jars from the cupboard. One is long and thin and one is wide.

“Which jar should I use?” Jillian asks her grandmother.

“Use whichever one will hold the most,” her grandmother says.

Jillian looks at the two jars. Here is what they look like.

Jillian has measured each jar to try to figure out which one will hold the most. She just isn’t sure what to do now.

First, let’s go back and reread the problem.

Jillian needs to figure out the volume of each cylinder. She can use the formula below to do this. Jillian suspects that the wide jar will hold more. What do you think

\begin{align*}V = \pi r^2h\end{align*}

Let’s start with the long thin jar. The diameter of the jar is 8 inches. We need the radius of the jar, so we can divide the diameter in half. The radius of this jar is 4 inches.

\begin{align*}V & = (3.14)4^2(16)\\ V & = (3.14)(16)(16)\\ V & = 803.84 \ in^ 3\end{align*}

Wow! That jar sure does hold a lot. Let’s work on the wide jar now. The diameter of this jar is 12 inches, so the radius is 6 inches.

\begin{align*}V & = (3.14)6^2(6)\\ V & = (3.14)(36)(6)\\ V & = 678.24 \ in^3\end{align*}

**Jillian is amazed. The long, thin jar holds more volume than the wide jar does. Jillian takes the beans and puts them into the jar.**

**Sometimes volume can be tricky! What looks like it holds more sometimes doesn’t!**

### Vocabulary

Here are the vocabulary words in this Concept.

- Volume
- the amount of space inside a three-dimensional figure.

- Surface Area
- the entire outer covering or surface of a three-dimensional figure. It is calculated by the sum of the areas of each of the faces and bases of a solid.

- Cylinder
- a three-dimensional figure with two congruent parallel circular bases and a curved flat surface connecting the bases.

- Radius
- the measure of the distance halfway across a circle.

### 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

Here is a video for review.

Khan Academy: Cylinder Volume and Surface Area

### Practice

Directions: Find the volume of each of the following cylinders.

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*}

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Cylinder

A cylinder is a solid figure with two parallel congruent circular bases.Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.Volume

Volume is the amount of space inside the bounds of a three-dimensional object.### Image Attributions

Here you'll learn to use formulas to find the volume of cylinders.

## Concept Nodes:

Cylinder

A cylinder is a solid figure with two parallel congruent circular bases.Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.Volume

Volume is the amount of space inside the bounds of a three-dimensional object.