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.

\begin{align*}V = Bh\end{align*}

\begin{align*}B\end{align*}

So, first figure out the area of the base.

\begin{align*}A = 3 \times 4 = 12\end{align*}

Next, multiply \begin{align*}B\end{align*}

\begin{align*}\begin{array}{rcl} h &=& 5 \\ V &=& 12 \times 5 = 60 \end{array}\end{align*}

The volume is 60 cubic inches or \begin{align*}in^3\end{align*}

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

\begin{align*}V = Bh\end{align*}

First, figure the area of the base.

The area of the base is \begin{align*}2 \times 8 = 16\end{align*}

Next, multiply the \begin{align*}B\end{align*}

\begin{align*}\begin{array}{rcl} V &=& 16 \times 3 \\ V &=& 48 \ in^3 \end{array}\end{align*}

The volume of this rectangular prism is \begin{align*}48 \ in^3\end{align*}

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 (\begin{align*}B\end{align*}

\begin{align*}V = Bh\end{align*}

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

\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2} (15 \times 6) \\ A &=& \frac{1}{2} (90) \\ A &=& 45 \text{ sq. units} \end{array}\end{align*}

Next, multiply this by the height.

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

The volume of the prism is \begin{align*}90 \ cm^3\end{align*}.

### Examples

#### Example 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.

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

\begin{align*}V = Bh\end{align*}

First, substitute in the given values. Remember, \begin{align*}B\end{align*} is length times width.

\begin{align*}\begin{array}{rcl} V &=& (8 \times 6)(1) \\ V &=& 48 \ ft^3 \end{array}\end{align*}

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.

\begin{align*}48 \div 2 = 24\end{align*}

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

#### Example 2

Find the volume of the prism.

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

\begin{align*}V = Bh\end{align*}

First, substitute in the given values. Remember \begin{align*}B\end{align*} is length times width.

\begin{align*}\begin{array}{rcl} V &=& (16 \times 9)(4) \\ V &=& 576 \ cm^3 \end{array}\end{align*}

The answer is \begin{align*}576 \ cm^3\end{align*}.

#### Example 3

Find the volume of the prism.

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

\begin{align*}V = Bh\end{align*}

Next, substitute in the given values. Remember \begin{align*}B\end{align*} is length times width. This is a square cube, so the length, width, and height are the same.

\begin{align*}\begin{array}{rcl} V &=& (5 \times 5)(5) \\ V &=& 125\ in^3 \end{array}\end{align*}

The answer is \begin{align*}125 \ in^3\end{align*}.

#### Example 4

Find the volume of the prism.

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

\begin{align*}V = Bh\end{align*}

First, substitute in the given values. Remember, \begin{align*}B\end{align*} is length times width.

\begin{align*}\begin{array}{rcl} V &=& (30 \times 5)(3) \\ V &=& 450\ in^3 \end{array}\end{align*}

The answer is \begin{align*}450 \ in^3\end{align*}.

#### Example 5

Find the volume of the prism.

To find the volume of a triangular prism, multiply the area of the base (\begin{align*}B\end{align*}) with the height of the prism.

\begin{align*}V = Bh\end{align*}

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

\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2}bh \\ A &=& \frac{1}{2} (5 \times 7) \\ A &=& \frac{1}{2} (35) \\ A &=& 17.5 \ sq. \ m \end{array}\end{align*}

Next, multiply this by the height.

\begin{align*}\begin{array}{rcl} V &=& Bh \\ V &=& (17.5)1 \\ V &=& 17.5 \ cm^3 \end{array}\end{align*}

The volume of the prism is \begin{align*}17.5 \ cm^3\end{align*}.

### 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 \begin{align*}h\end{align*} means the height of the triangular base and means the height of the whole prism.

- \begin{align*}b = 6 \ in, \ h = 4 \ in, \ H = 5 \ in\end{align*}
- \begin{align*}b = 7 \ in, \ h = 5 \ in, \ H = 9 \ in\end{align*}
- \begin{align*}b = 10 \ m, \ h = 8 \ m, \ H = 9 \ m\end{align*}
- \begin{align*}b = 12 \ m, \ h = 10 \ m, \ H = 13 \ m\end{align*}
- \begin{align*}b = 8 \ cm, \ h = 6 \ cm, \ H = 9 \ cm\end{align*}

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.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.16.