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# Heights of Cylinders Given Surface Area

## h = SA/(2πr) - r

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Heights of Cylinders Given Surface Area
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Josh is shopping for a koozie that is tall enough to keep a tall can of his favorite beverage, ice tea, cold.  A koozie is a styrofoam container into which a can of beverage slides to keep the can insulated and, thus, the beverage cold.  Josh finds a variety of koozies that are the same radius, but different heights.  He isn't sure how tall his ice tea can is.  He does, however, know the radius and surface area of the can.  The radius of the can is 1.5 inches and the surface area is 80 square inches.  What is the height of his can of ice tea?

In this concept, you will learn how to calculate the height of a cylinder given the radius and surface area.

### Finding the Height of a Cylinder Given Surface Area

Sometimes you are given the surface area of a cylinder and need to find its height. To do this, you simply fill the information you know into the surface area formula for a cylinder. So instead of solving for \begin{align*}SA\end{align*}, you solve for \begin{align*}h\end{align*}, the height of the cylinder.

Let's look at an example.

The surface area of a cylinder with a radius of 3 inches is \begin{align*}78 \pi\end{align*} square inches. What is the height of the cylinder?

First, substitute the values into the formula for surface area of a cylinder and then solve for height.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 78 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\ 78 \pi & = 2 \pi (9) + 6 \pi h\\ 78 \pi & = 18 \pi + 6 \pi h\\ 78 \pi - 18 \pi & = 6 \pi h \qquad \qquad \qquad \text{Subtract} \ 18 \pi \ \text{from both sides.}\\ 60 \pi & = 6 \pi h\\ 60 \pi \div 6 \pi & = h \qquad \qquad \quad \qquad \text{Divide both sides by} \ 6 \pi.\\ 10 \ in. & = h \end{align*}

The answer is 10 inches.

You can check your work by putting the height into the formula and solving for surface area.

\begin{align*}SA & = 2 \pi (3^2) + 2 \pi (3) (10)\\ SA & = 2 \pi (9) + 2 \pi (30)\\ SA & = 18 \pi + 60 \pi\\ SA & = 78 \pi\end{align*}

This is the surface area given in the problem, so the answer is correct.

Let’s look at another example.

Candice is going to use decoupage to decorate the outside of a cylindrical canister. The canister has a radius of 3 inches and has a surface area of \begin{align*}207.24 \ in^2\end{align*}. What is the height of the canister?

To figure this out, first convert the surface area value into a value times pi. This is done by dividing the surface area of 207.25 square inches by the value of pi, 3.14:

\begin{align*}\frac{207.25}{3.14}= 66\ (rounded)\end{align*}

Then, plug in the values for radius and surface area, replacing the surface area value of 207.25 with \begin{align*}66 \pi\end{align*}:

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 66 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\ 66 \pi & = 2 \pi (9) + 6 \pi h\\ 66 \pi & = 18 \pi + 6 \pi h\\ 66 \pi - 18 \pi & = 6 \pi h \qquad \qquad \qquad \text{Subtract} \ 18 \pi \ \text{from both sides.}\\ 66 \pi & = 6 \pi h\\ 66 \pi \div 6 \pi & = h \qquad \qquad \quad \qquad \text{Divide both sides by} \ 6 \pi.\\ 8 \ in. & = h \end{align*}

The answer is the height of the cylinder is \begin{align*}8 \ in\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Josh and his koozies.

What should the height of the koozie be to make sure that his can of tea fits completely into it if the surface area of the can is 80 square inches and the radius  is 1.5 inches?

First, calculate the surface area as multiplied by pi so we can plug the surface area and radius values into the surface area formula for a cylinder:

\begin{align*}\frac{80\ {in}^{2}}{3.14}\ = 25.5\end{align*}\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 25.5 \pi & = 2 \pi (1.5^2) + 2 \pi (1.5) h \end{align*}A48π=2πr2+2πrh=2π(32)+2π(3)h\begin{align*} \end{align*}

Next, complete the multiplication in the equation and subtract 4.5 pi from both sides::

\begin{align*}25.5 \pi & = 2 \pi (1.5^2) + 2 \pi (1.5) h\\ 25.5 \pi & = 2 \pi (2.25) + 3 \pi h\\ 25.5 \pi & = 4.5 \pi + 3 \pi h\\ \end{align*}  \begin{align*}25 \pi - 4.5 \pi & = 3 \pi h\\ 20.5 \pi & = 3 \pi h\\ \end{align*}

Then, divide both sides by \begin{align*}3 \pi\end{align*}, remembering to include the unit of measurement:\begin{align*}20.5 \pi & = 3 \pi h\\ 20.5 \pi \div 3 \pi & = h \\ 6.8 \ in. & = h \end{align*}

The answer is the height of the ice tea can is 6.8 inches.  So Josh should buy a koozie with that height to make sure the can is covered and his tea stays cold.

#### Example 2

Find the height of a cylinder with the following measurements:  \begin{align*}SA = 48 \pi, \ r = 3 \ in\end{align*}.

First, plug the surface area and radius values into the surface area formula for a cylinder and complete the multiplication:

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 48 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\ \end{align*}

\begin{align*}48 \pi & = 2 \pi (9) + 6 \pi h\\ 48 \pi & = 18 \pi + 6 \pi h\\ \end{align*}

Next, subtract \begin{align*}18 \pi\end{align*} from both sides:

\begin{align*}48 \pi & = 18 \pi + 6 \pi h\\ 48 \pi - 18 \pi & = 6 \pi h\\ 30 \pi & = 6 \pi h\\ \end{align*}

Then, divide both sides by  \begin{align*}6 \pi\end{align*} for the answer,  remembering to include the unit of measurement:

\begin{align*}30 \pi & = 6 \pi h\\ 30 \pi \div 6 \pi & = h \\ 5 \ in. & = h \end{align*}The answer is the height of the cylinder is 5 inches.

#### Example 3

Find the height of a cylinder with the following measurements:  \begin{align*}SA = 60 \pi, \ r = 2 \ in\end{align*}

First, plug the surface area and radius values into the surface area formula for a cylinder and complete the multiplication:

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 60 \pi & = 2 \pi (2^2) + 2 \pi (2) h\\ \end{align*}
\begin{align*}60 \pi & = 2 \pi (4) + 4 \pi h\\ 60 \pi & = 8 \pi + 4 \pi h\\ \end{align*}

Next, subtract \begin{align*}8 \pi\end{align*}from both sides:

\begin{align*}60 \pi & = 8 \pi + 4 \pi h\\ 60 \pi - 8 \pi & = 4 \pi h\\ 52 \pi & = 4 \pi h\\ \end{align*}

Then, divide both sides by \begin{align*}4 \pi\end{align*}, remembering to include the unit of measurement:

\begin{align*}52 \pi & = 4 \pi h\\ 52 \pi \div 4 \pi & = h \\ 13 \ in. & = h \end{align*}The answer is the height of the cylinder is 13 inches.

#### Example 4

Find the height of a cylinder with the following measurements:  \begin{align*}SA = 80 \pi, \ r = 4 \ m\end{align*}

First, plug the surface area and radius values into the surface area formula for a cylinder and complete the mutliplication:

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 80 \pi & = 2 \pi (4^2) + 2 \pi (4) h\\ \end{align*}
\begin{align*}80 \pi & = 2 \pi (16) + 8 \pi h\\ 80 \pi & = 32 \pi + 8 \pi h\\ \end{align*}

Then, subtract \begin{align*}32 \pi\end{align*}from both sides:

\begin{align*}80 \pi & = 32 \pi + 8 \pi h\\ 80 \pi -32 \pi & =8 \pi h\\ 48 \pi & = 8 \pi h\\ \end{align*}

Last, divide both sides by \begin{align*}8 \pi\end{align*}, remembering to include the unit of measurement:

\begin{align*}48 \pi & = 8 \pi h\\ 48 \pi \div 8 \pi & = h \\ 6 \ m. & = h \end{align*}

The answer is the height of the cylinder is 6 meters.

#### Example 5

Mrs. Javitz bought a container of oatmeal in the shape of a cylinder. If the container has a radius of 5 cm and a surface area of 534 square cm., what is its height?

First, calculate the surface area as multiplied by pi so you can plug the surface area and radius values into the surface area formula for a cylinder:

\begin{align*}\frac{534\ {cm}^{2}}{3.14}\ = 170\end{align*}

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 170 \pi & = 2 \pi (5^2) + 2 \pi (5) h \end{align*}

A48π=2πr2+2πrh=2π(32)+2π(3)h\begin{align*} \end{align*}Next, complete the multiplication in the equation and subtract 50 pi from both sides:

\begin{align*}170 \pi & = 2 \pi (5^2) + 2 \pi (5) h\\ 170 \pi & = 2 \pi (25) + 10 \pi h\\ 170 \pi & = 50 \pi + 10 \pi h\\ \end{align*}
\begin{align*}170 \pi - 50 \pi & = 10 \pi h\\ 120 \pi & = 10 \pi h\\ \end{align*}Then, divide both sides by \begin{align*}10 \pi\end{align*}, remembering to include the unit of measurement:

\begin{align*}120 \pi & = 10 \pi h\\ 120 \pi \div 10 \pi & = h \\ 12 \ cm. & = h \end{align*}

The answer is the oatmeal container has a height of 12 centimeters.

### Review

Find the height of each cylinder given the surface area and one other dimension.

1. \begin{align*}r = 5 \ in\end{align*}\begin{align*}SA = 376.8 \ in^2\end{align*}
2. \begin{align*}r = 6 \ in\end{align*}\begin{align*}SA = 527.52 \ in^2\end{align*}
3. \begin{align*}r = 5.5 \ in\end{align*}\begin{align*}SA = 336.765 \ in^2\end{align*}
4. \begin{align*}r = 4 \ cm\end{align*}\begin{align*}SA = 251.2 \ cm^2\end{align*}
5. \begin{align*}r = 5 \ m\end{align*}\begin{align*}SA = 471 \ m^2\end{align*}
6. \begin{align*}r = 2 \ cm\end{align*}\begin{align*}SA = 87.92 \ cm^2\end{align*}
7. \begin{align*}r = 2 \ cm\end{align*}\begin{align*}SA = 113.04 \ cm^2\end{align*}
8. \begin{align*}d = 4 \ m\end{align*}\begin{align*}SA = 125.6 \ m^2\end{align*}
9. \begin{align*}d = 8 \ cm\end{align*}\begin{align*}SA = 904.32 \ cm^2\end{align*}
10. \begin{align*}d = 6 \ cm\end{align*}\begin{align*}SA = 395.64 \ cm^2\end{align*}
11. \begin{align*}r = 3 \ cm\end{align*}\begin{align*}SA = 226.08 \ cm^2\end{align*}
12. \begin{align*}r = 14 \ cm\end{align*}\begin{align*}SA = 2637.6 \ cm^2\end{align*}
13. \begin{align*}r = 18 \ cm\end{align*}\begin{align*}SA = 4295.52 \ cm^2\end{align*}
14. \begin{align*}r = 12 \ cm\end{align*}\begin{align*}SA = 1959.36 \ cm^2\end{align*}
15. \begin{align*}r = 13 \ cm\end{align*}\begin{align*}SA = 3428.88 \ cm^2\end{align*}

To see the Review answers, open this PDF file and look for section 10.7.

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### Vocabulary Language: English

Cylinder

A cylinder is a solid figure with two parallel congruent circular bases.

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.