# 9.12: Volume of Prisms

- Page ID
- 6221

Use the formula \(V = Bh\)

Karen and Lance are building a backyard playground for their two children. They have a swing set, a merry-go-round, and a sandbox already set up. Lance is going to the store to buy sand for the sandbox and needs to know how much sand it can hold. The sandbox is 6 feet wide, 8 feet long, and 1 feet deep. How can Lance use a formula to calculate the total volume of this sandbox?

In this concept, you will learn to find volumes of rectangular and triangular prisms using formulas.

### Finding Volume of Prisms

**Volume** is the amount of space inside a solid figure.

Filling solid figures with cubes is a simple, easy way to understand volume. If you can count the cubes, you can figure out the volume. However, sometimes you will have to figure out the volume of a prism when there aren’t any cubes drawn inside it.

Take a look at the prism below.

This **rectangular prism** has a height of 5 inches, a width of 3 inches, and a length of 4 inches.

Here is a formula for finding the volume of this type prism.

\(V=Bh\)

B means the area of the base, which is the length times the width, and h means the height.

So, first figure out the area of the base.

\(A=3\times 4=12\)

Next, multiply \(B\) by \(h\).

\(\begin{aligned} h&=5\\ V&=12\times 5=60\end{aligned}\)

The volume is 60 cubic inches or \(in^{3}\). Remember, volume is in cubic units.

Let’s look at another example. Find the volume using the volume formula.

\(V=Bh\)

First, figure the area of the base.

The area of the base is \(2\times 8=16\)

Next, multiply the B by h. The height is 3 inches.

\(\begin{aligned} V&=16\times 3 \\ V&=48\: \text{ in}^{3}\end{aligned}\)

The volume of this rectangular prism is \(48\:\text{ in}^{3}\).

You can use the same formula for finding the volume of the **triangular prism**. Except this time, the area of the base is a triangle and not a rectangle.

Take a look at the triangular prism below.

To find the volume of a triangular prism, multiply the area of the base (\(B\)) with the height of the prism.

\(V=Bh\)

First, find the area of the triangular base using the formula for area of a triangle.

\(\begin{aligned} A&=\dfrac{1}{2}bh \\ A&=\dfrac{1}{2}(15\times 6) \\ A&=\dfrac{1}{2}(90) \\ A&=45\:\text{ sq. units}\end{aligned}\)

Next, multiply this by the height.

\(\begin{aligned} V&=Bh \\ V&=(45)h \\ V&=45(2) \\ V&=90\text{ cubic centimeters} or \text{ cm}^{3}\end{aligned}\)

The volume of the prism is \(90\text{ cm}^{3}\).

Example \(\PageIndex{1}\)

Earlier, you were given a problem about Lance and Karen’s sandbox.

The sandbox is 6 feet wide, 8 feet long, and 1 feet deep. Lance needs to know the volume.

**Solution**

To find the volume of the sandbox, which is a prism with one of its bases removed, use the following formula.

\(V=Bh\)

First, substitute in the given values. Remember, B is length times width.

\(\begin{aligned} V&=(8\times 6)(1)\\ V&=48\text{ ft}^{3}\end{aligned}\)

This is the total volume.

Next, to find out how much sand he needs to fill the sandbox halfway, divide the total volume by 2.

\(48\divide 2=24\)

Lance needs 24 cubic feet of sand to fill the sandbox halfway.

Example \(\PageIndex{2}\)

Find the volume of the prism.

**Solution**

To find the volume of a prism, use the following formula.

\(V=Bh\)

First, substitute in the given values. Remember B is length times width.

\(\begin{aligned} V&=(16\times 9)(4) \\ V&=576\text{ cm}^{3}\end{aligned}\)

The answer is \(576\text{ cm}^{3}\).

Example \(\PageIndex{3}\)

Find the volume of the prism.

**Solution**

First, to find the volume of a prism, use the following formula.

\(V=Bh\)

Next, substitute in the given values. Remember B is length times width. This is a square cube, so the length, width, and height are the same.

\(\begin{aligned} V&=(5\times 5)(5) \\V&=125\text{ in}^{3}\end{aligned}\)

The answer is \(125\text{ in}^{3}\).

Example \(\PageIndex{4}\)

Find the volume of the prism.

**Solution**

First, to find the volume of a prism, use the following formula.

\(V=Bh\)

First, substitute in the given values. Remember, B is length times width.

\(\begin{aligned} V&=(30\times 5)(3) \\ V&=450\text{ in}^{3}\end{aligned}\)

The answer is \(450\text{ in}^{3}\).

Example \(\PageIndex{5}\)

Find the volume of the prism.

**Solution**

To find the volume of a triangular prism, multiply the area of the base (B) with the height of the prism.

\(V=Bh\)

First, find the area of the triangular base using the formula for area of a triangle.

\(\begin{aligned} A&=\dfrac{1}{2}bh \\ A&=\dfrac{1}{2}(5\times 7) \\ A&=\dfrac{1}{2}(35) \\ A&=17.5 \text{ sq. m} \end{aligned}\)

Next, multiply this by the height.

\(\begin{aligned} V&=Bh \\ V&=(17.5)1 \\ V&=17.5\text{ cm}^{3}\end{aligned}\)

The volume of the prism is \(17.5\text{ cm}^{3}\).

## Review

Find the volume of each rectangular prism. Remember to label your answer in cubic units.

- Length = 5 in, width = 3 in, height = 4 in
- Length = 7 m, width = 6 m, height = 5 m
- Length = 8 cm, width = 4 cm, height = 9 cm
- Length = 8 cm, width = 4 cm, height = 12 cm
- Length = 10 ft, width = 5 ft, height = 6 ft
- Length = 9 m, width = 8 m, height = 11 m
- Length = 5.5 in, width = 3 in, height = 5 in
- Length = 6.6 cm, width = 5 cm, height = 7 cm
- Length = 7 ft, width = 4 ft, height = 6 ft
- Length = 15 m, width = 8 m, height = 10 m

Find the volume of each triangular prism. Remember that h\) means the height of the triangular base and H\) means the height of the whole prism.

- \(b=6\text{ in}\), \(h=4\text{ in}\), \(H=5\text{ in}\)
- \(b=7\text{ in}\), \(h=5\text{ in}\), \(H=9\text{ in}\)
- \(b=10\text{ m}\), \(h=8\text{ m}\), \(H=9\text{ m}\)
- \(b=12\text{ m}\), \(h=10\text{ m}\), \(H=13\text{ m}\)
- \(b=8\text{ cm}\), \(h=6\text{ cm}\), \(H=9\text{ cm}\)

Answer true or false for each of the following questions.

- Volume is the amount of space that a figure can hold inside it.
- The volume of a rectangular prism is always greater than the volume of a cube.
- The volume of a triangular prism is less than a rectangular prism with the same size base.
- A painter would need to know the
**surface area**of a house to do his/her job correctly. - If Marcus is covering his book with a book cover, Marcus is covering the surface area of the book.

## Vocabulary

Term | Definition |
---|---|

Net |
A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid. |

Rectangular Prism |
A rectangular prism is a prism made up of two rectangular bases and four rectangular faces. |

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

Triangular Prism |
A triangular prism is a prism made up of two triangular bases and three rectangular faces. |

## Additional Resources

Interactive Element

Video: Solid Geometry Volume

Practice: Volume of Prisms