# 10.7: Surface Area and Volume of Cylinders

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

## Introduction

*The Bean Containers*

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 lesson 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.**

*What You Will Learn*

By the end of this lesson you will be able to demonstrate the following skills:

- Identify surface area of cylinders as the sum of the areas of faces using nets.
- Find surface area of cylinders using formulas.
- Identify volume of cylinders as the sum of volumes of layers of unit cubes.
- Find volumes of cylinders using formulas.

*Teaching Time*

I. **Identify Surface Area of Cylinders as the Sum of the Areas of Faces Using Nets**

In the last lesson, you learned about how to calculate the ** surface area** and volume of different prisms. In this section, you will learn about calculating the surface area and volume of cylinders. Let’s start with calculating the surface area of a cylinder.

**Here is a cylinder. Notice that it has two parallel congruent circular bases. The face of the cylinder is one large rectangle. In fact, if you were to “unwrap” a cylinder here is what you would see.**

This is what the ** net** of a cylinder looks like.

**Just like when we were working with prisms, we can use the net of a cylinder to calculate the surface area of the cylinder.**

**How can we calculate the surface area of a cylinder using a net?**

To calculate the surface area of a cylinder using a net, we need to figure out the area of the two circles and the area of the rectangle too.

**Let’s think back to how to find the area of a circle**. To find the area of a circle, we use the following formula.

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

There are two circular bases in the cylinder, so we can multiply the area of the circle by two and have the sum of the two areas.

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

**The radius of the circles in the net above is 3 inches. We can substitute this given value into the formula and figure out the area of the two circles.**

\begin{align*}A & = 2\pi r^2\\ A & = 2(3.14)(3^2)\\ A & = 2(3.14)(9)\\ A & = 2(28.26)\\ A & = 56.52 \ in^2\end{align*}

**Next, we need to figure out the area of the curved surface. If you look at the net, the curved surface of the cylinder is rectangular in shape.**

**The length of the rectangle is the same as the circumference of the circle. Huh? Let’s look at the net. Since the length of the rectangle wraps around the circle rim, it is the same length as the circumference of the circle. To find the area of the curved surface, we need the circumference times the height.**

\begin{align*}A & = 2\pi rh\\ A & = 2(3.14)(3)(5)\\ A & = 2(3.14)(15)\\ A & = 2(47.1)\\ A & = 94.2 \ in^2\end{align*}

**Now we can add up the areas.**

\begin{align*}SA & = 56.52 + 94.2 = 150.72 \ in^2\end{align*}

**The surface area of the cylinder is \begin{align*}150.72 \ in^2\end{align*}.**

II. **Find Surface Areas of Cylinders Using Formula**

**Looking at the work that we just did, can we combine these steps to write a formula?**

You can see that we have gotten a “thumbs up” on this idea! Let's write a formula and use that formula to find the surface area of a cylinder.

**The formula for finding the surface area of a cylinder combines the formula for the area of the top and bottom circles with the formula for finding the area of the rectangular 'wrap' around the side. Remember that the wrap has a length equal to the circumference of the circular end, and a width equal to the height of the cylinder.** Here it is.

Let’s apply this formula with an example.

Example

**We work this problem through by substituting the given values into the formula**. 4 centimeters is the radius of the circular bases. 8 centimeters is the height of the cylinder.

\begin{align*}SA & = 2\pi r^2 + 2\pi rh\\ SA & = 2(3.14)(4^2) + 2(3.14)(4)(8)\\ SA & = 2(3.14)(16) + 2(3.14)(32)\\ SA & = 2(50.24) + 2(100.48)\\ SA & = 100.48 + 200.96\\ SA & = 301.44 \ cm^2\end{align*}

**The surface area of the cylinder is \begin{align*}301.44 \ cm^2\end{align*}**. Notice that this works well whether you have a net or a picture of a cylinder. As long as you use the formula and the given values, you can figure out the surface area of the cylinder.

**Practice a few of these on your own. Find the surface area of each cylinder using a formula.**

*Take a few minutes to check your work with a friend. Did you notice something in number 3?*

III. **Identify Volume of Cylinders as the Sum of Volumes of Layers of Unit Cubes**

Now that you have learned how to figure out the surface area of a cylinder, let’s look at ** volume**.

**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.

**How can we think about the volume of a 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. Here is an example.

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.

**One thing is sure though, no matter which formula we use, the volume of a cylinder is calculated in cubic units.**

IV. **Find Volumes of Cylinders Using Formulas**

**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*}

Now let’s apply this formula when working with an example.

Example

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.**

- \begin{align*}d = 10 \ ft, \ h = 12 \ ft\end{align*}
- \begin{align*}r = 6 \ in, \ h = 10 \ in\end{align*}

*Take a few minutes to check your work with a partner.*

## Real Life Example Completed

*The Bean Containers*

**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. Underline any important information in red.**

**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 sometimes doesn’t!**

## Vocabulary

- 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.

## Technology Integration

Khan Academy, Cylinder Volume and Surface Area

Other Videos:

- http://www.mathplayground.com/mv_volume_cylinders.html – This is a video by Brightstorm about how to find the volume of a cylinder.

- http://www.mathplayground.com/mv_surface_area_cylinders.html – This is a Brightstorm video on how to calculate the surface area of a cylinder.

## Time to Practice

Directions: Calculate the surface area of each of the following cylinders using nets.

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Directions: Find the surface area and volume of each of the following cylinders.

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