# 9.12: Volume of Prisms

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Use the formula $$V = Bh$$ Figure $$\PageIndex{1}$$

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. Figure $$\PageIndex{2}$$

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. Figure $$\PageIndex{3}$$

$$V=Bh$$

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. Figure $$\PageIndex{4}$$

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. Figure $$\PageIndex{5}$$

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. Figure $$\PageIndex{6}$$

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. Figure $$\PageIndex{7}$$

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. Figure $$\PageIndex{8}$$

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.

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

1. $$b=6\text{ in}$$, $$h=4\text{ in}$$, $$H=5\text{ in}$$
2. $$b=7\text{ in}$$, $$h=5\text{ in}$$, $$H=9\text{ in}$$
3. $$b=10\text{ m}$$, $$h=8\text{ m}$$, $$H=9\text{ m}$$
4. $$b=12\text{ m}$$, $$h=10\text{ m}$$, $$H=13\text{ m}$$
5. $$b=8\text{ cm}$$, $$h=6\text{ cm}$$, $$H=9\text{ cm}$$

Answer true or false for each of the following questions.

1. Volume is the amount of space that a figure can hold inside it.
2. The volume of a rectangular prism is always greater than the volume of a cube.
3. The volume of a triangular prism is less than a rectangular prism with the same size base.
4. A painter would need to know the surface area of a house to do his/her job correctly.
5. 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.