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Volume of Cylinders

V = πr^2h

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Volume of Cylinders


Alicia went to the store to buy three cans of soup. Each can has a diameter of 5.4 inches and a height of 6.7 inches. What is the total volume of soup that Alicia bought?

In this concept, you will learn to calculate the volume of different cylinders.

Volume

The volume of a solid figure is the measure of how much three-dimensional space it takes up or holds. Volume is measured in cubic units, because you are multiplying three dimensions.

Let’s look at the formula for the volume of a cylinder\begin{align*}V = \pi r^2 h\end{align*}

All you need to know is the radius of the circular faces and the height of the cylinder. You simply put these numbers into the formula and solve.

Let’s try an example.

Find the volume of the cylinder below.

First, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ V &=& \pi (5.5)^2 \times 2.7 \end{array}\end{align*}

Next, use algebra to solve for the volume.

\begin{align*}\begin{array}{rcl} V &=& \pi (5.5)^2 \times 2.7 \\ V &=& \pi (30.25) \times 2.7 \\ V &=& 256.6 \end{array}\end{align*}

The answer is 256.6.

The volume is \begin{align*}256.6 \ cm^3\end{align*}.

You can also think about working backwards if you have the volume and one other dimension. Then, you can problem solve to figure out the missing dimension.

Let’s look at an example.

A cylinder with a radius of 3 inches has a volume of \begin{align*}140.4 \ \pi\end{align*} cubic inches. What is the height of the cylinder?

First, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ 104.4 \pi &=& \pi (3)^2 \times h \\ \end{array}\end{align*}

Next, use algebra to solve for the height, \begin{align*}h\end{align*}.

\begin{align*}\begin{array}{rcl} 140.4 \pi &=& \pi(3)^2 \times h \\ 140.4 \pi &=& \pi(9) \times h \\ \frac{140.4 \pi}{9 \pi} &=& \frac{9 \pi \times h}{9 \pi} \\ h &=& 15.6 \end{array}\end{align*}

The answer is 15.6.

The height is 15 in.

Examples

Example 1

Earlier, you were given a problem about Alicia and her soup?

Alicia wants to know the volume of the three cans of soup if each has a diameter of 5.4 inches and a height of 6.7 inches.

First, find the radius of the cylinder. Remember the radius is half the diameter.

\begin{align*}\begin{array}{rcl} r &=& \frac{d}{2} \\ r &=& \frac{5.4}{2} \\ r &=& 2.7 \end{array}\end{align*}

Next, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ V &=& \pi (2.7)^2 \times 6.7 \end{array}\end{align*}

Next, use algebra to solve for the volume.

\begin{align*}\begin{array}{rcl} V &=& \pi (2.7)^2 \times 6.7 \\ V &=& \pi (7.29) \times 6.7 \\ V &=& 153.44 \end{array} \end{align*}
Then, find the total volume for the 3 cans.

\begin{align*}\begin{array}{rcl} V &=& 3 \times 153.44 \\ V &=& 460.32 \end{array}\end{align*}

The answer is 460.32.

Alicia bought \begin{align*}460.32 \ in^3\end{align*} of soup.

Example 2

A water tank has a radius of 45 feet and a height of 300 feet. How many cubic feet of water will the tank hold when it is full?

First, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ V &=& \pi (45)^2 \times 300 \\ \end{array}\end{align*}

Next, use algebra to solve for the volume.

\begin{align*}\begin{array}{rcl} V &=& \pi (45)^2 \times 300 \\ V &=& \pi (2025) \times 300 \\ V &=& 1908517.5 \end{array}\end{align*}

The answer is 1,908,517.5.

The volume is \begin{align*}1,908,517.5 \ ft^3\end{align*}.

Example 3

Find the volume of a cylinder with a radius of 3 inches and a height of 9 inches.

First, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ V &=& \pi (3)^2 \times 9 \\ \end{array}\end{align*}

Next, use algebra to solve for the volume.

\begin{align*}\begin{array}{rcl} V &=& \pi (3)^2 \times 9 \\ V &=& \pi (9) \times 9 \\ V &=& 254.47 \end{array}\end{align*}

The answer is 254.47.

The volume is \begin{align*}254.47 \ in^3\end{align*}.

Example 4

Find the volume of a cylinder with a radius of 2.5 feet and a height of 6 feet.

First, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ V &=& \pi (2.5)^2 \times 6 \\ \end{array}\end{align*}

Next, use algebra to solve for the volume.

\begin{align*}\begin{array}{rcl} V &=& \pi (2.5)^2 \times 6 \\ V &=& \pi (6.25) \times 6 \\ V &=& 117.81 \end{array}\end{align*}

The answer is 117.81.

The volume is \begin{align*}117.81 \ ft^3\end{align*}.

Example 5

Find the volume of a cylinder with a diameter of 5 inches and a height of 7 inches.

First, find the radius of the cylinder. Remember the radius is half the diameter.

\begin{align*}\begin{array}{rcl} r &=& \frac{d}{2} \\ r &=& \frac{5}{2} \\ r &=& 2.5 \end{array}\end{align*}

Next, substitute what you know into the formula for the volume of a cylinder.

\begin{align*}\begin{array}{rcl} V &=& \pi r^2 h \\ V &=& \pi(2.5)^2 \times 7 \end{array}\end{align*}

Then, use algebra to solve for the volume.

\begin{align*}\begin{array}{rcl} V &=& \pi(2.5)^2 \times 7 \\ V &=& \pi(6.25) \times 7 \\ V &=& 137.44 \end{array}\end{align*}

The answer is 137.44.

The volume is \begin{align*}137.44 \ in^3\end{align*}.

Review

Answer the questions below the diagram.

1. Name the figure pictured above.

2. What is the diameter of the figure?

3. What is the radius is the figure?

4. What is the volume of the figure?

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

5. \begin{align*}r = 4 \ \text{feet}, h = 5 \ \text{feet}\end{align*}

6. \begin{align*}r = 6 \ cm, h = 8 \ cm\end{align*}

7. \begin{align*}r = 4.5 \ \text{feet}, h = 5 \ \text{feet}\end{align*}

8. \begin{align*} r = 3.5 \ m, h = 7 \ m\end{align*}

9. \begin{align*}r = 13 \ \text{ft}, h = 2 \ \text{ft}\end{align*}

10. \begin{align*}r = 11 \ m, h = 12 \ m\end{align*}

11. \begin{align*} r = 1.5 \ \text{ft}, h = 3 \ \text{ft}\end{align*}

12. \begin{align*}r = 7 \ \text{in}, h = 12 \ \text{in}\end{align*}

13. \begin{align*}r = 8 \ cm, h = 11 \ cm\end{align*}

14. \begin{align*}r = 5 \ m, h = 9 \ m\end{align*}

15. \begin{align*}r = 4.5 \ \text{ft}, h = 6 \ \text{ft}\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.11.  

Resources

 

Vocabulary

Cylinder

Cylinder

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

Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.
Surface Area

Surface Area

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

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

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