# Surface Area of Cylinders

## SA = 2πr(h+r)

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Surface Area of Cylinders
Credit: Nic McPhee
Source: https://www.flickr.com/photos/nics_events/3049856021/in/photolist-5DviWv-8MFbTG-8MC7fD-8MC7o2-8MC71H-8MC7rx-8MC6RM-8MFbWS-8MFbLw-8MC6Qr-8MC77e-8MFc21-8MC6Ue-8MC73V-8MFcmb-9s3Jny-9s3Fbu-9s3KCS-9s3JMQ-9rZLJc-9rZKNk-9rZMwv-9s3Le1-9s3NrJ-9s3KWE-rJgjC-p7jumn-rJgjJ-8eu6yQ-rJgjy-rJgjv-rJgjA-9e85W5-9duoir-9dxs7W-9duowZ-9dxsrE-9rZNte-9rZNDk-9s35TN-9rZHw4-9s3F89-9s3Fpb-9rZLXx-9rZNc6-9s3Ga1-9s3FCw-9rZP6B-9rZNmn-9rZNZX

Tevin is in charge of prop design for the Drama Club’s upcoming production. The play will take place in Ancient Rome, so Tevin has to wrap two cylindrical canisters so that they resemble stone pillars. He needs to know the surface area of each canister so he can order the special adhesive covering he needs. Each canister is 12 inches in diameter and 30 inches tall. How can Tevin use this information to figure out the surface area of each canister?

In this concept, you will learn to identify the surface area of cylinders by using nets.

### Finding Surface Area of Cylinders

A cylinder has two parallel congruent circular bases. The face of the cylinder is one large rectangle.

If you were to “unwrap” a cylinder, here is what you would see.

This is what the net of a cylinder looks like. To calculate the surface area of a cylinder using a net, you need to figure out the area of the two circles and the area of the rectangle.

First, to find the area of a circle, use the following formula.

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

There are two circular bases in the cylinder, so multiply the area of the circle by two for 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. Substitute this given value into the formula and figure out the area of the two circles.

\begin{align*}\begin{array}{rcl} 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{array} \end{align*}

The area of the two circles is 56.52 square inches.

Next, 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. To find the area of the curved surface, you need to know the circumference (the length) and the height.

Use this formula to find the area of the rectangle.

\begin{align*}\begin{array}{rcl} 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{array}\end{align*}

\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. Here is the formula.

You can use this formula to find the surface area of the cylinder below.

First, substitute the given values into the formula.

The radius of the circular bases is 4 cm and the height of the cylinder is 8 cm.

\begin{align*}\begin{array}{rcl} 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{array}\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.

### Examples

#### Example 1

Earlier, you were given a problem about Tevin and the props he’s making for the Drama Club production.

Tevin has to create two stone pillars from cylindrical canisters and will be covering their surface area to make them look like stone. Each canister is 12 inches in diameter and 30 inches tall.

First, note the diameter is given instead of the radius, so divide the diameter by 2.

\begin{align*}12 \div 2 = 6\end{align*}

Next, substitute the given values into the formula.

\begin{align*} \begin{array}{rcl} SA &=& 2 \pi r^2 + 2 \pi r h \\ SA &=& 2(3.14)(6^2) + 2(3.14)(6)(30) \\ SA &=& 2(3.14)(36) + 2(3.14)(180) \\ SA &=& 226.08 + 1130.4 \\ SA &=& 1356.48 \ in^2 \end{array}\end{align*}

The surface area of each canister is \begin{align*}1356.48 \ in^2\end{align*}.

#### Example 2

Use the formula for finding the surface area of a cylinder to answer the following question.

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

First, substitute the given values into the formula then solve.

\begin{align*}\begin{array}{rcl} SA &=& 2 \pi r^2 + 2 \pi r h \\ 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{array}\end{align*}

The answer is \begin{align*}678.24 \ in^2\end{align*}.

#### Example 3

Find the surface area of the cylinder.

First, substitute the given values into the formula then solve.

\begin{align*}\begin{array}{rcl} SA &=& 2 \pi r^2 + 2 \pi r h \\ SA &=& 2(3.14)(2^2) + 2(3.14)(2)(12) \\ SA &=& 2(3.14)(4) + 2(3.14)(24) \\ SA &=& 25.12 + 150.72 \\ SA &=& 175.84 \ in^2 \end{array}\end{align*}

The answer is \begin{align*}175.84 \ in^2\end{align*}.

#### Example 4

Find the surface area of the cylinder.

First, substitute the given values into the formula then solve.

\begin{align*}\begin{array}{rcl} SA &=& 2 \pi r^2 + 2 \pi r h \\ SA &=& 2(3.14)(5^2) + 2(3.14)(5)(10) \\ SA &=& 2(3.14)(25) + 2(3.14)(50) \\ SA &=& 157 + 314 \\ SA &=& 471 \ m^2 \\ \end{array}\end{align*}

The answer is \begin{align*}471 \ m^2\end{align*}.

#### Example 5

Find the surface area of the cylinder.

First, note the diameter is given instead of the radius, so divide the diameter by 2.

\begin{align*}8 \div 2 = 4\end{align*}

Next, substitute the given values into the formula then solve.

\begin{align*}\begin{array}{rcl} SA &=& 2 \pi r^2 + 2 \pi r h \\ SA &=& 2(3.14)(4^2) + 2(3.14)(4)(12)\\ SA &=& 2(3.14)(16) + 2(3.14)(48)\\ SA &=& 100.48 + 301.44\\ SA &=& 402.24 \ in^2 \end{array}\end{align*}

The answer is \begin{align*}402.24 \ in^2\end{align*}.

### Review

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

[Figure9][Figure10][Figure11][Figure12][Figure13]

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

1. \begin{align*}r = 4 \ in, h = 8 \ in\end{align*}
2. \begin{align*}r = 5 \ in, h = 15 \ in\end{align*}
3. \begin{align*}r = 8 \ m, h = 16 \ m\end{align*}
4. \begin{align*}r = 11 \ m, h = 20 \ m\end{align*}
5.  \begin{align*}r = 3.5 \ m, h = 8 \ m\end{align*}
6. \begin{align*}d = 4 \ ft, h = 6 \ ft\end{align*}
7. \begin{align*}d = 10 \ ft, h = 15 \ ft\end{align*}
8. \begin{align*}d = 20 \ cm, h = 25 \ cm\end{align*}
9. \begin{align*}d = 18 \ in, h = 24 \ in\end{align*}
10.  \begin{align*}d = 20 \ ft, h = 45 \ ft\end{align*}

To see the Review answers, open this PDF file and look for section 10.17.

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