Grandma Marion is knitting toilet paper covers as presents for the upcoming holiday season. She needs to figure out how many balls of yarn she will need for each cover. In order to do this, she needs to know the total outside area of the toilet paper roll. The radius of the top, and base, of the toilet paper roll measures 2 inches, and the height of the roll is 5 inches. If one of Grandma Marion's balls of yarn can knit a piece that is 100 square inches, will one ball be enough to make a toilet paper cover?

In this concept, you will learn how to find the surface area of a cylinder.

### Finding the Surface Area of a Cylinder

A **cylinder** is a solid figure that has two circular faces at each end. The side of a cylinder is not called a face because it is curved. The **surface area** of a cylinder is the total of area of each circular face and the side of the cylinder. Imagine a can of soup. The top, bottom, and label around the can would make up the surface area of the can. To find the surface area, you must be able to calculate the area of each face and the side, and then add these areas together.

One way to calculate the surface area of a cylinder is to use a **net**. Imagine you could unroll the soup can so that it is completely flat. You would have something that looks like this.

The circles show the top and bottom faces of the cylinder, and the rectangle shows the side, as if it were unrolled. With the net, you can see each face of the cylinder more clearly.

To find the surface area, you need to calculate the area for each circle in the net. You use the formula \begin{align*}A = \pi r^2\end{align*}**radius** is the distance from the center point of a circle to any point on the **circumference**, or perimeter, of the circle. If you know the radius, you can calculate its area. Look closely again at the cylinder above. The two circular faces are congruent, so they must have the same radius and diameter. Let’s calculate the area for each face.

\begin{align*}&\text{bottom face} && \text{top face}\\
A &= \pi r^2 && A = \pi r^2\\
A &= \pi (4)^2 && A = \pi (4)^2\\
A &= 16 \pi && A = 16 \pi\\
A &= 16 \times\ 3.14 && A = 16\times\ 3.14\\
A &= 50.24 \ cm^2 && A = 50.24 \ cm^2\end{align*}

The area of each circular face is 50.24 square centimeters.

Now you need to find the area of the side. The net shows that when you “unroll” the cylinder, the side is actually a rectangle. Recall that the formula used to find the area of a rectangle is \begin{align*}A = lw\end{align*}

\begin{align*}C & = 2 \pi r\\
C & = 2 \pi 4\\
C & = 8 \pi\\
C & = 8 \times 3.14 = 25.12 \ cm\end{align*}

The circumference of the top and bottom of the cylinder, or the length of the side, is 25.12 centimeters.

To find the area of the cylinder's side, multiply the circumference of the circle, or length of the cylinder, by the height, or width, of the cylinder.

Now that you know the area of both circular faces and the side, add them together to find the surface area of the cylinder.

\begin{align*}& \text{bottom face} \qquad \text{top face} \qquad \quad \text{side} \qquad \qquad \quad \text{surface area}\\
& 50.24 \ cm^2 \quad + \ \ 50.24 \ cm^2 \ + \ 200.96 \ cm^2 \ = \ 301.44 \ cm^2\end{align*}

The answer is the total surface area of the cylinder is 301.44 square centimeters.

You can also use one formula to represent the surface area of the faces and side of a cylinder.

You may have noticed in the previous section that the two circular faces had the same area. This is because they are **congruent**, and thus have the same radius. You can therefore calculate the area of the pair of circular faces at one time. Simply double the area formula, which gives you **\begin{align*}2 \pi r^2\end{align*}.**

You can also combine the measurements for the side into a simpler equation. You need to find the circumference by using the formula \begin{align*}2 \pi r\end{align*}, and then multiply this by the height of the cylinder \begin{align*}h\end{align*}. The formula for the side of the cylinder then becomes **\begin{align*}2 \pi rh\end{align*}.**

When you combine the formulas for the faces and the side, you get this formula:

\begin{align*}SA = 2 \pi r^2 + 2 \pi rh\end{align*}

Now let’s apply this formula to the example from the previous section.

In this cylinder, \begin{align*}r = 4\end{align*} inches and \begin{align*}h = 8\end{align*} inches. Simply put these numbers into the formula and solve for surface area.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ SA & = 2 \pi (4)^2 + 2 \pi (4)(8)\\ SA & = 2 \pi (16) + 2 \pi (32)\\ SA & = 32 \times\ 3.14 + 64 \times\ 3.14\\ SA & = 100.48 +\ 200.96\\ SA & = 301.44 \ cm.^2\end{align*}

Again, the answer is the surface area of this cylinder is 301.44 square centimeters.

This formula just saves a little time. Let’s look at another example.

What is the surface area of the figure below?

You have all of the measurements you need. Let’s put them into the formula and solve for surface area, \begin{align*}SA\end{align*}.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ SA & = 2 \pi (3.5)^2 + 2 \pi (3.5) (28)\\ SA & = 2 \pi (12.25) + 2 \pi (98)\\ SA & = 24.5 \pi + 196 \pi\\ SA & = 220.5 \times\ 3.14\\ SA & = 692.37 \ cm^2\end{align*}

This cylinder has a surface area of 692.37 square centimeters.

### Examples

#### Example 1

Earlier, you were given a problem about Grandma Marion's toilet paper covers.

She wants to know how many balls of yarn are required for one cover. The dimensions of the toilet paper are height is 5 inches and radius is 2 inches. One ball of yarn will create a knitted piece of 100 square inches.

First, plug in the values of the radius and height into the surface area formula and multiply.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ SA & = 2 \pi (2)^2 + 2 \pi (2)(5)\\ SA & = 2 \pi (4) + 2 \pi (10)\\\end{align*}

\begin{align*}SA & = 2 \pi (4) + 2 \pi (10)\\ SA & = 8 \pi + 20 \pi\\ \end{align*}

Next, replace the value for pi and multiply.

\begin{align*}SA & =8 \times\ 3.14 + 20 \times\ 3.14\\ SA & = 25.12 +\ 62.8\\\end{align*}

Then, add the two values together for the answer, making sure to include the appropriate unit of measurement.

**\begin{align*}SA & = 25.12 +\ 62.8\\
SA & = 87.92\ {in}^{2}\end{align*}**

The answer is the surface area of the toilet paper roll is 87.92 square inches. Therefore, one ball of yarn is enough to knit a toilet paper cover.

#### Example 2

What is the surface area of the cylinder below using the net of the figure?

First, draw the net. This is done by drawing the bottom and top faces, each with a radius of 7 inches, and the side rectangle, which is 14 inches in length, between the circular faces.

Next, calculate the areas of the circles representing the top and bottom of the cylinder, and the rectangle representing the side of the cylinder.

Areas of circles:

\begin{align*}& \text{bottom face} && \text{top face}\\ A &= \pi r^2 && A = \pi r^2\\ A &= \pi (7)^2 && A = \pi (7)^2\\ A &= 49 \times\ 3.14 && A = 49 \times\ 3.14\\ A &= 153.86 \ in.^2 && A = 153.86 \ in.^2\end{align*}

Area of rectangle:

**\begin{align*}& \text{side}\\
C &= 2 \pi r\\
C &= 2 \pi (7)\\
C &= 14 \times\ 3.14\\
C &= 43.96 \ in.^2\\
\\
Ch &= 43.96 \times 14 = 615.44 \ in.^2\end{align*}**

Then, add these areas together to find the surface area of the cylinder, making sure to include the appropriate unit of measurement.

\begin{align*}153.86 \quad + \quad 153.86 \quad + \quad 615.44 \quad = \quad 923.16 \ in.^2\end{align*}

The answer is the surface area of the cylinder pictured above is 923.16 square inches.

Calculate the surface area of a cylinder with a radius of 7 in and a height of 12 in.

First, plug in the values of the radius and height into the surface area formula, then multiply.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ SA & = 2 \pi (7)^2 + 2 \pi (7)(12)\\ SA & = 2 \pi (49) + 2 \pi (84)\\\end{align*}

\begin{align*}SA & = 2 \pi (49) + 2 \pi (84)\\ SA & = 98 \pi + 168 \pi\\ \end{align*}

Next, replace the value for pi and multiply.

\begin{align*}SA & = 98 \times\ 3.14 + 168 \times\ 3.14\\ SA & = 307.72 +\ 527.52\\\end{align*}

Then, add the two values together for the surface area, making sure to include the unit of measurement.

**\begin{align*}SA & = 307.72 +\ 527.52\\
SA & = 835.24\ {in}^{2}\end{align*}**

The answer is the surface area of the cylinder is ** \begin{align*}835.24\end{align*} **square inches.

#### Example 3

Calculate the surface area of a cylinder with a radius of 5 ft and a height of 10 ft.

First, plug in the values of the radius and height into the surface area formula and multiply.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ SA & = 2 \pi (5)^2 + 2 \pi (5)(10)\\ SA & = 2 \pi (25) + 2 \pi (50)\\\end{align*}

\begin{align*}SA & = 2 \pi (25) + 2 \pi (50)\\
SA & = 50 \pi + 100 \pi\\
\end{align*}Next, replace the value for pi and multiply.

\begin{align*}SA & =50 \times\ 3.14 + 100 \times\ 3.14\\ SA & = 157 +\ 314\\\end{align*}

Then, add the two values together for the answer, making sure to include the appropriate unit of measurement.

**\begin{align*}SA & = 157 +\ 314\\
SA & = 471\ {ft}^{2}\end{align*} **

The answer is the surface area of the cylinder is \begin{align*}471\end{align*} square feet.

#### Example 4

Calculate the surface area of a cylinder with a radius of 7 in. and a height of 12 in.

First, plug in the values of the radius and height into the surface area formula and multiply.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ SA & = 2 \pi (7)^2 + 2 \pi (7)(12)\\ SA & = 2 \pi (49) + 2 \pi (84)\\\end{align*}

\begin{align*}SA & = 2 \pi (49) + 2 \pi (84)\\ SA & = 98 \pi + 168 \pi\\ \end{align*}

Next, replace the value for pi and multiply.

\begin{align*}SA & =98 \times\ 3.14 + 168 \times\ 3.14\\ SA & = 307.72 +\ 527.52\\\end{align*}

Then, add the two values together for the answer, making sure to include the appropriate unit of measurement.

**\begin{align*}SA & = 307.72 +\ 527.52\\
SA & = 835.24\ {in}^{2}\end{align*} **

The answer is the surface area of the cylinder is \begin{align*}835.24\end{align*} square inches.

### Review

Find the surface area of each cylinder given its height and radius.

- \begin{align*}r = 1 \ m, \ h = 3 \ m\end{align*}
- \begin{align*}r = 2 \ cm, \ h = 4 \ cm\end{align*}
- \begin{align*}r = 6 \ in, \ h = 10 \ in\end{align*}
- \begin{align*}r = 4 \ in, \ h = 6 \ in\end{align*}
- \begin{align*}r = 5 \ in, \ h = 10 \ in\end{align*}
- \begin{align*}r = 8 \ ft, \ h = 6 \ ft\end{align*}
- \begin{align*}r = 10 \ m, \ h = 15 \ m\end{align*}
- \begin{align*}r = 9 \ cm, \ h = 12 \ cm\end{align*}
- \begin{align*}r = 6 \ m, \ h = 8 \ m\end{align*}
- \begin{align*}r = 2 \ cm, \ h = 3 \ cm\end{align*}

Find the surface area of each cylinder given its height and diameter.

- \begin{align*}d = 8 \ m, \ h = 11 \ m\end{align*}
- \begin{align*}d = 10 \ in, \ h = 14 \ in\end{align*}
- \begin{align*}d = 8 \ cm, \ h = 10 \ cm\end{align*}
- \begin{align*}d = 12 \ m, \ h = 15 \ m\end{align*}
- \begin{align*}d = 15 \ in, \ h = 20 \ in\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.6.