# 9.21: Heights of Cylinders Given Surface Area or Volume

Use formulas to find the height of a cylinder, given the volume or surface area.

Gregory's family just bought a hot tub for their lake house. The hot tub company told his family that the tub holds 125 cubic feet of water. Gregory is interested to know how deep the hot tub is. He measures the diameter of the top and finds that the hot tub is 6 feet across. What is the height, or depth, of the hot tub?

In this concept, you will learn how to calculate the height of a cylinder when given the volume and radius or diameter.

### Finding the Height of a Cylinder Given Volume

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

\begin{aligned} V&= \pi r^{2}h \\ V&= \pi (2)^{2}(10) \\ V&= \pi (4)(10) \\ V&= 40\pi \\ V&= 125.6 \text{ in}^{3}

Let's look at an example.

A cylinder with a radius of 2 inches has a volume of 125.6 cubic inches. What is the height of the cylinder?

The volume and the radius are given, so substitute these into the formula and then solve for h, the height.

\begin{aligned}V&= \pi r^{2}h \\ 125.6&=(3.14)(22)h \\ 125.6&=(3.14)(4)h \\ 125.6&=12.56h \\ 125.6&\divide 12.56 \\ 10 \text{ in}&=12.56 h\divide 12.56=h\end{aligned}

The height of the cylinder is 10 inches.

Check your work by substituting the answer in for the height. You should get a volume of 125.6 cubic inches.

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

What is the height of a cylinder that has a radius of 6 cm and a volume of 904.32 cubic cm?

Again, you have been given the volume and the radius. Put this information into the formula along with the value of pi and solve for h, the height.

\begin{aligned}V&= \pi r^{2}h \\ 904.32&=(3.14)(62)h \\ 904.32&=(3.14)(36)h \\ 904.32&=113.04 h \\ 904.32\divide 113.04&=113.04h\divide 113.04 \\ 8 \text{ cm}&=h\end{aligned}

The height of this cylinder is 8 centimeters.

Example $$\PageIndex{1}$$

Earlier, you were given a problem about Gregory and his family's hot tub.

To figure this out, use the formula for volume of a cylinder. He already knows the volume of the tub is 125 cubic feet and the diameter is 6 feet.

Solution

First, divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder.

\begin{aligned}r&= 6\divide 2 \\ r&= 3 \\ V&= \pi r^{2}h \\ 125&=(3.14)(3^{2})h\end{aligned}

Next, square the radius and multiply the values together.

\begin{aligned}125&=(3.14)(3^{2})h \\ 125&=(3.14)(9)h \\ 125&=28.26h\end{aligned}

Last, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement.

\begin{aligned}125&=28.26h \\ 125\divide 28.26&=28.26 h\divide 28.26 \\ 4.42 \text{ ft}&=hv\end{aligned}

The answer is Gregory's hot tub is 4.42 feet deep.

Example $$\PageIndex{2}$$

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?

Solution

First, divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder.

\begin{aligned}r&= 16\divide 2 \\ r&= 8 \\ V&=(3.14)(82)h \\ 2,110.08&=\pi r^{2}h\end{aligned}

Next, square the radius and multiply the values together.

\begin{aligned}2,110.08&=(3.14)(8^{2})h \\ 2,110.08&=(3.14)(64)h \\ 2,110.08&=200.96h\end{aligned}

Then, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement.

\begin{aligned}2,110.08&=200.96h \\ 2,110.08\divide 200.96&=200.96 h\divide 200.96 \\ 10.5 \text{ in}&=h\end{aligned}

The answer is Javier must make his tank 10.5 inches tall for his tank to hold 2,110.08 cubic inches of water.

Example $$\PageIndex{3}$$

Find the height of a cylinder with radius = 6 inches and volume = 904.32 cubic inches.

Solution

First, plug the values of the volume, pi, and radius into the formula for volume of a cylinder.

\begin{aligned}V&= \pi r^{2}h \\ 904.32&=(3.14)(6^{2})h\end{aligned}

Next, square the radius and multiply the values together.

\begin{aligned}904.32&=(3.14)(62)h \\ 904.32&=(3.14)(36)h \\ 904.32&=113.04h\end{aligned}

Last, divide each side by 113.04 for the answer, remembering to include the appropriate unit of measurement.

\begin{aligned}904.32&=113.04h \\ 904.32\divide 113.04&=113.04h\divide 113.04 \\ 8 \text{ in}&=h\end{aligned}

The answer is the height of the cylinder is 8 inches.

Example $$\PageIndex{4}$$

Find the height of a cylinder with radius = 3 meters and volume = 354.34 cubic meters.

Solution

First, plug the values of the volume, pi, and radius into the formula for volume of a cylinder.

\begin{aligned}V&= \pi r^{2} h \\ 354.34&=(3.14)(3^{2})h \end{aligned}

Next, square the radius and multiply the values together.

\begin{aligned}354.34&=(3.14)(3^{2})h \\ 904.32&=(3.14)(9)h \\ 354.34&=28.26h \end{aligned}

Last, divide each side by 28.26 for the answer, remembering to include the appropriate unit of measurement.

\begin{aligned}354.34v=28.26h \\ 354.34\divide 28.26&=28.26h\divide 28.26 \\ 9 \text{ m}&=h\end{aligned}

The answer is the height of the cylinder is 9 meters.

Example $$\PageIndex{5}$$

Find the height of a cylinder with radius = 5 feet and volume = 785 cubic feet.

Solution

First, plug the values of the volume, pi, and radius into the formula for volume of a cylinder.

\begin{aligned}V&= \pi r^{2} h \\ 785&=(3.14)(52)h\end{aligned}

Next, square the radius and multiply the values together.

\begin{aligned}785&=(3.14)(52)h \\ 785&=(3.14)(25)h \\ 785&=78.5h \end{aligned}

Last, divide each side by 78.5 for the answer, remembering to include the appropriate unit of measurement.

\begin{aligned}785&=78.5h \\ 785\divide 78.5&=78.5\divide 78.5 \\ 10 \text{ ft}&=h\end{aligned}

The answer is the height of the cylinder is 10 feet.

## Review

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

1. $$r= 6 \text{ in}$$, $$V&= 904.32 \text{ in}^{3}$$
2. $$r= 5 \text{ in}$$, $$V&= 706.5 \text{ in}^{3}$$
3. $$r= 7 \text{ ft}$$, $$V&= 2307.9 \text{ ft}^{3}$$
4. $$r= 8 \text{ ft}$$, $$V&= 4019.2 \text{ ft}^{3}$$
5. $$r= 7 \text{ ft}$$, $$V&= 1538.6 \text{ ft}^{3}$$
6. $$r= 12\text{ m}$$, $$V&= 6330.24\text{ m}^{3}$$
7. $$r= 9\text{ m}$$, $$V&= 4069.49\text{ m}^{3}$$
8. $$r= 10\text{ m}$$, $$V&= 5652\text{ m}^{3}$$
9. $$r= 12 \text{ in}$$, $$V&= 11304 \text{ in}^{3}$$
10. $$r= 11 \text{ ft}$$, $$V&= 3039.52 \text{ ft}^{3}$$
11. $$r= 10 \text{ in}$$, $$V&= 1570 \text{ in}^{3}$$
12. $$r= 9.5 \text{ in}$$, $$V&= 1700.31 \text{ in}^{3}$$
13. $$r= 8\text{ m}$$, $$V&= 1808.64\text{ m}^{3}$$
14. $$r= 14 \text{ ft}$$, $$V&= 5538.96 \text{ ft}^{3}$$
15. $$r= 13.5 \text{ in}$$, $$V&= 4005.85 \text{ in}^{3}$$

## Resources

Interactive Element

## Vocabulary

Term Definition
Cubic Units Cubic units are three-dimensional units of measure, as in the volume of a solid figure.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.

Interactive Element

Video: Cylinder Volume and Surface Area

Practice: Heights of Cylinders Given Surface Area or Volume

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