# 10.7: Heights of Cylinders Given Surface Area

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

What if you know the surface area of a cylinder, but don't know the height? Look at this dilemma.

Candice 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 Mom. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Candice's canister has a radius of 3 inches and has a surface area of \begin{align*}207.3 \ in^2\end{align*}.

What is the height of the canister?

**Use this Concept to learn about how to figure out the height of a cylinder given the radius and surface area.**

### Guidance

**Sometimes we may be given the surface area of a cylinder and need to find its height. We simply fill the information we know into the surface area formula. This time, instead of solving for \begin{align*}SA\end{align*}, we solve for \begin{align*}h\end{align*}, the height of the cylinder.**

Let’s look at how we can apply this information.

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?

**We know that the radius of the cylinder is 3 inches. We also know that the surface area of the cylinder is \begin{align*}78 \pi\end{align*} square inches. Sometimes a problem will include pi in the amount because it is more precise. We’ll see that this also makes our calculations easier. All we have to do is put \begin{align*}78 \pi\end{align*} into the formula in place of \begin{align*}SA\end{align*}. Then we can use the formula to solve for the height, \begin{align*}h\end{align*}.**

\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*}

**We used the formula to find that a cylinder with a radius of 3 inches and a surface area of \begin{align*}78 \pi\end{align*} has a height of 10 inches.**

We can check our 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 our answer is correct. Let’s look at another example.

Now try a few of these on your own.

Find the height in each problem.

#### Example A

\begin{align*}SA = 48 \pi, \ r = 3 \ in\end{align*}

**Solution:\begin{align*}2 \ in\end{align*}**

#### Example B

\begin{align*}SA = 60 \pi, \ r = 2 \ in\end{align*}

**Solution:\begin{align*}5 \ in\end{align*}**

#### Example C

\begin{align*}SA = 80 \pi, \ r = 4 \ m\end{align*}

**Solution: \begin{align*}2 \ m\end{align*}**

Here is the original problem once again.

Candice 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 Mom. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Candice's 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, let's use the formula.

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

### Vocabulary

Here are the vocabulary words in this Concept.

- Cylinder
- a three-dimensional figure with two circular bases.

- Surface Area
- the measurement of the outside of a three-dimensional figure.

- Net
- a two-dimensional representation of a three-dimensional figure.

### Guided Practice

Here is one for you to try on your own.

Mrs. Javitz bought a container of oatmeal in the shape of the cylinder. If the container has a radius of 5 cm and a surface area of \begin{align*}170 \pi\end{align*}, what is its height?

**Answer**

Once again, we have been given the surface area as a function of pi. That’s no problem we simply put the whole number, \begin{align*}170 \pi\end{align*}, in for \begin{align*}SA\end{align*} in the formula. We also know that the container has a radius of 5 centimeters, so we can use the formula to solve for \begin{align*}h\end{align*}, the height of the container.

\begin{align*}SA & = 2 \pi r^2 + 2 \pi rh\\ 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\\ 170 \pi - 50 \pi & = 10 \pi h \quad \qquad \qquad \text{Subtract} \ 50 \pi \ \text{from both sides.}\\ 120 \pi & = 10 \pi h\\ 120 \pi \div 10 \pi & = h \qquad \qquad \qquad \ \ \text{Divide both sides by} \ 10 \pi.\\ 12 \ cm & = h\end{align*}

**The oatmeal container must have a height of 12 centimeters.**

### Video Review

Here is a video for review.

Khan Academy, Cylinder Volume and Surface Area

### Practice

Directions: 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 = 336.8255 \ 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 = 439.6 \ m^2\end{align*}

6. \begin{align*}r = 2 \ m\end{align*}

\begin{align*}SA = 87.92 \ cm^2\end{align*}

7. \begin{align*}r = 2 \ cm\end{align*}

\begin{align*}SA = 75.35 \ 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 = 351.68 \ cm^2\end{align*}

10. \begin{align*}d = 6 \ cm\end{align*}

\begin{align*}SA = 282.6 \ 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 = 4295.53 \ 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 = 3347.24 \ cm^2\end{align*}

### Image Attributions

Here you'll learn to find the heights of cylinders given surface area and one other dimension.