# 10.13: Heights of Cylinders Given Volume

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

**Practice**Heights of Cylinders Given Volume

Have you ever been in a pool that had an interesting shape?

The cylindrical pool in Trevor’s backyard holds 15,700 cubic feet of water. If the diameter of the pool is 50 feet, how deep is the pool?

**To figure this out, you will need to know how to use the volume of a cylinder to find another dimension. Use this Concept and you will know how to figure out the solution to this dilemma.**

### Guidance

Sometimes, you will know the volume of a cylinder and you won’t know the height of it. Think about a water tower that is cylindrical in shape. We might know how much volume the tank will hold, but not the height of it. When this happens, we can use our formula to find the missing height of the cylinder.

A cylinder with a radius of 2 inches has a volume of125.6 cubic inches. What is the height of the cylinder?

**What is the problem asking us to find? We need to solve for the height of the cylinder. The problem tells us the volume and the radius. We put these into the formula and then solve for \begin{align*}h\end{align*}, the height.**

\begin{align*}V & = \pi r^2h\\ 125.6 & = \pi (2)^2 h\\ 125.6 & = 4 \pi h\\ 125.6 & = 12.56 \ h\\ 125.6 \div 12.56 & = h\\ 10 \ in. & = h\end{align*}

**The height of the cylinder is 10 inches.**

**We used the volume formula to solve for \begin{align*}h\end{align*} and found that the height of the cylinder is 10 inches. We can check our work by putting this number in for the height. We should get a volume of 125.6 cubic inches.**

\begin{align*}V & = \pi r^2h\\ V & = \pi (2)^2 (10)\\ V & = \pi (4) (10)\\ V & = 40 \pi\\ V & = 125.6 \ in.^3\end{align*}

Our calculation was correct! Let’s look at another example.

What is the height of a cylinder that has a radius of 6 cm and a volume of \begin{align*}904.32 \ cm^3\end{align*}?

**Again, we have been given the volume and the radius. We put this information into the formula and solve for \begin{align*}h\end{align*}, the height.**

\begin{align*}V & = \pi r^2h\\ 904.32 & = \pi (6)^2 h\\ 904.32 & = 36 \pi h\\ 904.32 & = 113.04 \ h\\ 904.32 \div 113.04 & = h\\ 8 \ cm & = h\end{align*}

**The height of this cylinder is 8 centimeters.**

Find the height of each cylinder given the radius and volume.

#### Example A

\begin{align*}r = 6 \ in, \ \text{Volume} = 904.32 \ in^3\end{align*}

**Solution: \begin{align*}h = 8 \ in\end{align*}**

#### Example B

\begin{align*}r = 3 \ m, \ \text{Volume} = 254.34 \ m^3\end{align*}

**Solution: \begin{align*}h = 9 \ m\end{align*}**

#### Example C

\begin{align*}r = 5 \ ft, \ \text{Volume} = 785 \ ft^3\end{align*}

**Solution: \begin{align*}h = 10 \ ft\end{align*}**

Here is the original problem once again.

The cylindrical pool in Trevor’s backyard holds 15,700 cubic feet of water. If the diameter of the pool is 50 feet, how deep is the pool?

To figure this out, we use the formula for finding the volume of a cylinder.

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

Next, we fill in the given information. Notice we have been given a diameter of 50 feet. We need to radius, so we divide the diameter in half. While the problem is asking for depth, the height would be the depth of the pool because the pool is cylindrical in shape.

\begin{align*}15,700 = (3.14)(25^2)h\end{align*}

\begin{align*}15,700 = 1962.5h\end{align*}

\begin{align*}15,700 \div 1962.5 = h\end{align*}

\begin{align*}8 = h\end{align*}

**The depth of the pool is eight feet.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Volume
- the amount of space contained inside a three-dimensional solid. Volume is often used to measure capacity or liquid.

- Cubic Units
- how we measure volume. It is measured in cubic units because we multiply the length, the width and the height of a solid.

- Cylinder
- a solid with two circular bases and a flat rounded side.

### Guided Practice

Javier wants to construct a cylindrical container to hold enough water for his pet fish. He read that the fish needs to live in 2,110.08 cubic inches of water. If he constructs a tank with a diameter of 16 inches, how tall must he make it so that it holds the right amount of water?

Again, the first thing we need to do is decide what the problem is asking us to find. We need to know how tall Javier must make the tank. In other words, we need to find the height of the cylinder.

What information have we been given? We know that the tank must hold 2,110.08 cubic inches of water. This is the volume. What about the radius? We know that the diameter of the tank will be 16 inches, so the radius will be \begin{align*}16 \div 2 = 8\end{align*} inches. We put the radius and volume into the volume formula and solve for \begin{align*}h\end{align*}, the height.

\begin{align*}V & = \pi r^2h\\ 2,110.08 & = \pi (8)^2 h\\ 2,110.08 & = 64 \pi h\\ 2,110.08 & = 200.96h\\ 2,110.08 \div 200.96 & = h\\ 10.5 \ in. & = h\end{align*}

**In order for his tank to hold 2,110.08 cubic inches of water, Javier must make his tank 10.5 inches tall.**

### Video Review

Here is a video for review.

- This is a Khan Academy video on the volume of cylinders.

### Practice

Directions: Given the volume and the radius, find the height of each cylinder.

1. \begin{align*}r = 6 \ in, \ V = 904.32 \ in^3\end{align*}

2. \begin{align*}r = 5 \ in, \ V = 706.5 \ in^3\end{align*}

3. \begin{align*}r = 7 \ ft, \ V = 2307.9 \ ft^3\end{align*}

4. \begin{align*}r = 8 \ ft, \ V = 4019.2 \ ft^3\end{align*}

5. \begin{align*}r = 7 \ ft, \ V = 1538.6 \ ft^3\end{align*}

6. \begin{align*}r = 12 \ m, \ V = 6330.24 \ m^3\end{align*}

7. \begin{align*}r = 9 \ m, \ V = 4069.49 \ m^3\end{align*}

8. \begin{align*}r = 10 \ m, \ V = 5652 \ m^3\end{align*}

9. \begin{align*}r = 12 \ in, \ V = 11304 \ in^3\end{align*}

10. \begin{align*}r = 11 \ ft, \ V = 3039.52 \ ft^3\end{align*}

11. \begin{align*}r = 10 \ in, \ V = 1570 \ in^3\end{align*}

12. \begin{align*}r = 9.5 \ in, \ V = 1700.31 \ in^3\end{align*}

13. \begin{align*}r = 8 \ m, \ V = 1808.64 \ m^3\end{align*}

14. \begin{align*}r = 14 \ ft, \ V = 5538.96 \ ft^3\end{align*}

15. \begin{align*}r = 13.5 \ in, \ V = 4005.85 \ in^3\end{align*}

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