Have you ever tried to wrap a cylinder? Take a look at this dilemma.

**Mrs. Johnson is wrapping a cylindrical package in brown paper so that she can mail it to her son. The package is 22 centimeters tall and 11 centimeters across. How much paper will she need to cover the package?**

**You will learn how to solve problems like this one in this Concept.**

### Guidance

**A three–dimensional or solid figure has length, width and depth.** It is not simply a flat two–dimensional plane figure.

**What is a cylinder?**

A cylinder has two parallel congruent circular bases with a curved rectangle as its side.

**One of the things that we can measure when working with three–dimensional figures is called** *surface area***. Surface area is the total of the areas of each face of a solid figure. Imagine you could wrap one of the figures above in wrapping paper, like a present.**

The amount of wrapping paper needed to cover the figure represents its surface area. To find the surface area, we must be able to calculate the area of each face and then add these areas together.

There are several different ways to calculate surface area.

One way is to use a ** net**.

**A net is a two-dimensional diagram of a three-dimensional figure.** Imagine you could unfold a box so that it is completely flat. You would have something that looks like this.

We can look at the net of a cylinder too.

With the net of a cylinder, we would need to calculate the area of each circle and the area of the curved side of the cylinder. Then we could add these values together to find the surface area.

How can we find the surface area of a cylinder?

To find the surface area, we need to calculate the area for each circle in the net. We use the formula \begin{align*}A =\pi r^2\end{align*} to find the area of a circle. If we know the radius or diameter of each circle, we 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*}& \mathbf{bottom \ face} && \mathbf{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=50.24 \ cm^2 && A=50.24 \ cm^2\end{align*}

**The area of each circular face is 50.24 square centimeters when we use 3.14 to approximate pi.**

Now we need to find the area of the side. The net shows us that, when we “unroll” the cylinder, the side is actually a rectangle. Recall that the formula we use to find the area of a rectangle is \begin{align*}A = lw\end{align*}. For cylinders, the width of the rectangle is the same as the height of the cylinder. In this case, the height of the cylinder is 8 centimeters.

What about the length? The length is actually the same as the perimeter of the circle, which we call its circumference. When we “roll” up the side, it fits exactly once around the circle. To find the area of the cylinder’s side, then, we multiply the circumference of the circle by the height of the cylinder. We find the circumference of a circle with the formula \begin{align*}C = 2 \pi r\end{align*}, and then we multiply it by the height. Let’s try it.

\begin{align*}C &=2 \pi r\\ C &=2 \pi 4\\ C &=8 \pi\\ C &=25.12 \times 8 = 200.96 \ cm^2\end{align*}

**Now we know the area of both circular faces and the side. Let’s add them together to find the surface area of the cylinder.**

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

**The total surface area of the cylinder is 301.44 square centimeters. Using a net helped us to locate the faces and the find the measurements of the side.**

Did you know that you can use a formula to find the surface area of a cylinder?

We use the formula, \begin{align*}Ph + 2B\end{align*}, to find the surface area of cylinders, too. As we know, the top and bottom faces of a cylinder are circles. The perimeter of a circle is its radius. We find circumference by using the formula \begin{align*}2 \pi r\end{align*}. Then we multiply it by the height of the cylinder.

To find the area of the base, \begin{align*}B\end{align*}, we use the area formula for circles: \begin{align*}\pi {r^2}\end{align*}. We still multiply it by 2 because there is a circular top face and bottom face. This gives us the formula

\begin{align*}& SA =2 \pi {r^2} + 2 \pi rh\\ & \qquad \ \ (2B) \quad \ (Ph)\end{align*}

**This formula may look long and intimidating, but all we need to do is put in the values for the radius of the circular faces and the height of the cylinder and solve.**

*Write this formula and the sentence on how to use it down in your notebooks.*

Now let's apply this information.

**What is the surface area of the figure below? Use 3.14 to approximate pi.**

**We have all of the measurements we 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 \pi \\ SA &= 692.37 \ {cm^2}\end{align*}

**This cylinder has a surface area of 692.37 square centimeters.**

That wasn’t so bad! We just have to be careful to put each measurement in the right place in the formula and take it one step at a time.

**Sometimes, we can have a cylinder that has been cut. We call it a** *truncated cylinder***. This is where you only see a section of the cylinder and will need to figure out the surface area of what you see.**

**Now let’s say that one – half of the cylinder is pictured. We know that the radius of the circle will not change, so we can use that given measurement. The height of the cylinder will change because it has been cut in half. Therefore, we can figure out the surface area by using the given measurements and the same formula. We don’t have to do anything different because we are looking for the measurement of what is shown, the surface area of half of the cylinder.**

\begin{align*}SA &=2 \pi {r^2} + 2 \pi rh\\ SA &= 2 \pi (2^2) + 2 \pi (2) (4)\\ SA &= 2 \pi (4) + 2 \pi (8)\\ SA &=8 \pi + 16 \pi\\ SA &=75.36 \ {cm^2}\end{align*}

**We could also find the surface area of the whole cylinder by changing the height from 4 cm to 8 cm. Then we could use the same measurement for radius and calculate the surface area of the cylinder.**

Find the surface area of each cylinder.

#### Example A

A cylinder with a radius of 6 inches and a height of 5 inches.

**Solution: \begin{align*}414.48\end{align*} square inches**

#### Example B

A cylinder with a radius of 4 cm and a height of 12 cm.

**Solution: \begin{align*}401.92\end{align*} square cm**

#### Example C

A cylinder with a diameter of 10 meters and a height of 15 meters.

**Solution: \begin{align*}1099\end{align*} square meters**

Now let's go back to the dilemma from the beginning of the Concept.

**The picture clearly shows us the height and diameter of the cylinder, so let’s use the formula for finding the surface area. But be careful—we have been given the diameter, not the radius. We need to divide it by 2 to find the radius: \begin{align*}11 \div 2 = 5.5\end{align*}. Now we have the radius and height, so we can put these in for the appropriate variables in the formula.**

\begin{align*}SA &= 2 \pi r^2 + 2 \pi rh\\ SA &= 2 \pi (5.5^2) +2 \pi (5.5)(22)\\ SA &= 2 \pi (30.25) + 2 \pi (121)\\ SA &= 60.5 \pi + 242 \pi \\ SA &= 302.5 \pi\\ SA &= 949.85 \ cm^2\end{align*}

**Mrs. Johnson will need 949.85 square centimeters of brown paper in order to wrap the entire package.**

### Vocabulary

- Three Dimensional Figures
- solid figures that have length, width and height.

- Prisms
- three – dimensional figures with parallel, congruent polygons as bases and rectangular side faces.

- Cylinders
- three – dimensional figures with circular parallel congruent bases and a curved rectangle as the side.

- Surface Area
- the measurement of the outer covering on a solid figure.

- Net
- the pattern of a solid figure-what a solid figure would look like if it were drawn out as a pattern.

- Truncated Cylinder
- a cylinder that is cut in part from a complete cylinder.

### Guided Practice

Here is one for you to try on your own.

**What is the surface area of the figure below?**

**Solution**

Look carefully at the cylinder. This time we have been given the diameter, not the radius. Remember, the diameter of a circle is always twice the length of the radius. We can divide the diameter’s length by 2 to find the radius: \begin{align*}13 \div 2 = 6.5\end{align*}. Now we have the radius and the height, so let’s put the numbers into the formula and solve.

\begin{align*}SA &= 2 \pi r^2 + 2 \pi rh\\ SA &= 2 \pi (6.5^2) + 2 \pi (6.5) (11)\\ SA &= 2 \pi (42.25) + 2 \pi (71.5)\\ SA &= 84.6 \pi + 143 \pi\\ SA &= 227.6 \pi\\ SA &= 714.66 \ {ft^2}\end{align*}

**This cylinder has a surface area of 714.66 square feet when we approximate pi as 3.14.**

### Video Review

### Practice

- What is the name of this figure?
- What is the shape of the base of this figure?
- How many bases are there?
- What is the surface area of this figure?
- Which measurement is needed the radius or the diameter?

- What is the name of this figure?
- Which measurement is given the radius or the diameter?
- What is the surface area of the figure?

Directions: Use what you have learned to answer each question.

- A cylindrical water tank is 35 long and 10 feet across. How much sheet metal is the tank made of?
- Did you use area or surface area to solve this problem?
- True or false. You can only find the surface area if you know the volume.
- True or false. Surface area and volume measure the same thing.
- True or false. Surface area measures the outside of a cylinder.
- True or false. You need the radius to find the surface area of a cylinder.
- True or false. The radius is one-half of the diameter.