Charlotte wants to know how much of her favorite morning beverage, hot chocolate, fills her favorite mug. She measures the mug and finds the height is 4 inches and the diameter of the top is 3.5 inches. What is the volume of hot chocolate that Charlotte's mug holds?

In this concept, you will learn how to calculate the volume of a cylinder.

### Finding the Volume of a Cylinder

A **cylinder** is a solid shape that exists in three-dimensional space. A cylinder has two faces that are circles. You do not call the side of a cylinder a face because it is curved.

The **volume** of a cylinder is the measure of how much three-dimensional space it takes up or holds. Imagine a thermos. If you fill the thermos with water, the volume of the thermos is the amount of water it will hold. You measure volume in **cubic units**, because you are multiplying three dimensions: length, width, and height. The width of the thermos is the same as the diameter of the circular face.

You know that the two bases are circular, so you are going to need to know the area of the circle to figure out how much space can be contained on top of it. A cylinder is tall, or has height, so you are going to need to know the height of the cylinder.

First, let’s think about the area of the circular bases. To find the area of a circle, use the following formula where A = area, \begin{align*}\pi\end{align*}(pi) is a constant equal to 3.14 when rounded, and r = radius which is half the diameter:

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

But you also need the height \begin{align*}(h)\end{align*} of the cylinder to figure into this calculation. If you put all of these parts together, you have the following formula.

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

Let's look at an example.

Find the volume of a cylinder with a radius of 5 cm and a height of 7 cm.

Start by substituting the values of the cylinder into the formula.

\begin{align*}V & = \pi r^2 h \\ V & = (3.14)(5^2 )(7) \\ V & =(3.14)(25)(7) \\ V & = 549.5 \ cm^3 \end{align*}

The answer is the volume of this cylinder is 549.5 cubic centimeters.

Let's look at another example.

Find the volume of a cylinder with a diameter of 12 inches and a height of 8 inches.

First, notice that we have been given the diameter and not the radius. You can divide the diameter in half which will give the radius of the circular base of the cylinder, in this case, 6 inches.

Now substitute the values into the formula and solve for the volume.

\begin{align*}V & = \pi r^2 h \\ V & = (3.14)(36)(8) \\ V & = 904.32 \ in^3 \end{align*}

The answer is the volume of the cylinder is 904.32 cubic inches.

### Examples

#### Example 1

Earlier, you were given a problem about Charlotte's hot chocolate mug.

The mug is 4 inches in height with a diameter of 3.5 inches and Charlotte wants to know the volume.

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

\begin{align*}r&\ =\ 3.5 \div\ 2\\ r&\ =\ 1.75\end{align*}

\begin{align*}V & = \pi r^2 h \\ V & = (3.14)(1.75^2 )(4) \\ \end{align*}

Next, square the radius.\begin{align*}V & = (3.14)(1.75^2 )(4) \\ V & =(3.14)(3.06)(4) \\\end{align*}

Then, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.

\begin{align*}V & =(3.14)(3.06)(4) \\ V & = 38.43\ {in}^{3}\end{align*}

The answer is the volume of Charlotte's mug is 38.43 cubic inches. Her mug will hold 38.43 cubic inches of hot chocolate.

#### Example 2

Find the volume of a cylinder with a radius of 3 inches and height of 7 inches.

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

\begin{align*}V & = \pi r^2 h \\ V & = (3.14)(3^2 )(7) \\ \end{align*}

Next, square the radius.

\begin{align*}V & = (3.14)(3^2 )(7) \\ V & =(3.14)(9)(7) \\\end{align*}

Then, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.

\begin{align*}V & =(3.14)(9)(7) \\ V & = 197.82\ {in}^{3}\end{align*}

The answer is the volume of the cylinder is** \begin{align*}197.82 \ in^3\end{align*}.**

#### Example 3

Find the volume of a cylinder with radius = 2.5 mm and height = 4 mm.

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

\begin{align*}V & = \pi r^2 h \\ V & = (3.14)(2.5^2 )(4) \\ \end{align*}

Then, square the radius.\begin{align*}V & = (3.14)(2.5^2 )(4) \\ V & =(3.14)(6.25)(4) \\\end{align*}

Next, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.

\begin{align*}V & =(3.14)(6.25)(4) \\ V & = 78.5\ {mm}^{3}\end{align*}

The answer is the volume of the cylinder is** \begin{align*}78.5 \ mm^3\end{align*}.**

#### Example 4

Find the volume of a cylinder with diameter = 14 in and height = 9 in.

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

\begin{align*}r& =\ 14\ \div\ 2\\ r& =\ 7\end{align*}

\begin{align*}V & = \pi r^2 h \\ V & = (3.14)(7^2 )(9) \\ \end{align*}

Next, square the radius.\begin{align*}V & = (3.14)(7^2 )(9) \\ V & =(3.14)(49)(9) \\\end{align*}

Then, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.

\begin{align*}V & =(3.14)(49)(9) \\ V & = 1,384.74\ {in}^{3}\end{align*}

The answer is the volume of the cylinders 1,384.74 cubic inches.

### Review

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

- \begin{align*}r = 5 \ in, \ h = 8 \ in\end{align*}
- \begin{align*}r = 4 \ in, \ h = 7 \ in\end{align*}
- \begin{align*}r = 3 \ ft, \ h = 5 \ ft\end{align*}
- \begin{align*}r = 3 \ ft, \ h = 8 \ ft\end{align*}
- \begin{align*}r = 4 \ cm, \ h = 9 \ cm\end{align*}
- \begin{align*}r = 6 \ m, \ h = 12 \ m\end{align*}
- \begin{align*}r = 7 \ in, \ h = 14 \ in\end{align*}
- \begin{align*}r = 5 \ m, \ h = 10 \ m\end{align*}
- \begin{align*}r = 2 \ m, \ h = 11 \ m\end{align*}
- \begin{align*}r = 3 \ cm, \ h = 12 \ cm\end{align*}
- \begin{align*}r = 6 \ cm, \ h = 11 \ cm\end{align*}
- \begin{align*}r = 4 \ m, \ h = 14 \ m\end{align*}
- \begin{align*}r = 13 \ cm, \ h = 26 \ cm\end{align*}
- \begin{align*}r = 8 \ in, \ h = 14 \ in\end{align*}
- \begin{align*}r = 4.5 \ cm, \ h = 16.5 \ cm\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.12.