# 10.17: Surface Area of Cylinders

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

**Practice**Surface Area of Cylinders

Do you know what modge podge is? Have you ever used decoupage of some kind?

Jillian is going to use modge podge to decorate the outside of a cylindrical canister. She wants to make it into a decorative pencil holder as a gift for her grandmother. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Jillian's canister is 6 inches in diameter and 8 inches tall.

How much surface area will she need to cover?

**To figure this out, you will need to know how to calculate the surface area of a cylinder. This Concept will teach you what you need to know to accomplish this task.**

### Guidance

You have 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*}.**

**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.

**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.

Now it's time for you to try a few. Find the surface area of each cylinder.

#### Example A

**Solution: \begin{align*}175.84\end{align*} sq. in.**

#### Example B

**Solution: \begin{align*}471\end{align*} sq. m**

#### Example C

**Solution:\begin{align*}703.36\end{align*} sq. in.**

Here is the original problem once again.

Jillian is going to use modge podge to decorate the outside of a cylindrical canister. She wants to make it into a decorative pencil holder as a gift for her grandmother. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Jillian's canister is 6 inches in diameter and 8 inches tall.

How much surface area will she need to cover?

To solve this problem of surface area, we can use the formula for finding the surface area of a cylinder. Then we substitute in the given values and solve.

\begin{align*}SA & = 2\pi r^2 + 2\pi rh\\ SA & = 2(3.14)(3^2) + 2(3.14)(3)(8)\\ SA & = 2(3.14)(9) + 2(3.14)(24)\\ SA & = 2(28.29) + 2(75.36)\\ SA & =56.58 + 150.72\\ SA & = 207.3 \ in^2\end{align*}

**This is the surface area of Jillian's cylinder.**

### Vocabulary

Here are the vocabulary words in this Concept.

- 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.

What is the surface area of a cylinder with a radius of 6 inches and a height of 12 inches?

**Answer**

To complete this problem, we can use the formula for surface area presented in the Concept and then substitute in the given values.

\begin{align*}SA & = 2\pi r^2 + 2\pi rh\\ SA & = 2(3.14)(6^2) + 2(3.14)(6)(12)\\ SA & = 2(3.14)(36) + 2(3.14)(72)\\ SA & = 226.08 + 452.16\\ SA & = 678.24 \ in^2\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

Khan Academy, Cylinder Volume and Surface Area

### Practice

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

1.

2.

3.

4.

5.

Directions: Calculate the surface area of the following cylinders given these dimensions.

6. r = 4 in, h = 8 in

7. r = 5 in, h = 15 in

8. r = 8 m, h = 16 m

9. r = 11 m, h = 20 m

10. r = 3.5 m, h = 8 m

11. d = 4 ft, h = 6 ft

12. d = 10 ft, h = 15 ft

13. d = 20 cm, h = 25 cm

14. d = 18 in, h = 24 in

15. d = 20 ft, h = 45 ft

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### Image Attributions

Here you'll learn to identify the surface area of cylinders by using nets.