# 7.5: Volume of Prisms

**At Grade**Created by: CK-12

## Learning Objectives

- Find the volume of a prism.

## Volume of a Right Rectangular Prism

**Volume** is a measure of how much space a 3-dimensional figure occupies.

This means that the **volume** tells you how much a 3-dimensional figure can hold.

*What does volume represent?*

*Volume is the space inside a 3-dimensional solid.*

*One way to understand volume is to compare it to surface area.*

*We can use real-life examples of objects to compare volume to surface area:*

*A fish tank:*

*Surface area = the glass used to build the outside of the tank*

*Volume = the water inside the tank*

*A pillow:*

*Surface area = the fabric used to make the pillowcase*

*Volume = the feathers or stuffing inside the pillow*

*Can you think of other examples?*

The basic unit of volume is the **cubic** unit — cubic centimeter, cubic inch, cubic meter, cubic foot, and so on. Each basic cubic unit has a measure of 1 for its length, width, and height:

______________________ is the measure of space inside a solid object.

The basic unit of volume is a ________________________ unit.

In calculating volume, it is important to know that if 2 polyhedrons (or solids) are *congruent*, then their **volumes** are *congruent* also.

A **right rectangular prism** is a prism with *rectangular* **bases** and the *angle* between each base and its rectangular lateral sides is also a *right angle*. You can recognize a right rectangular prism by its “box” shape, like in the diagram below.

The **volume** of a solid is the *sum* of the volumes of all of its non-overlapping parts. Using this, we can find the volume of a **right rectangular prism** by counting boxes.

The box below measures 2 units in height, 4 units in width, and 3 units in depth. Each layer has (\begin{align*}2 \cdot 4\end{align*}) cubes or 8 cubes.

Together, you get 3 groups of (\begin{align*}2 \cdot 4\end{align*}) so the total volume is:

\begin{align*}V & = 2 \cdot 4 \cdot 3\\ &= 24\end{align*}

The volume is 24 cubic units.

This same pattern is true for *any* **right rectangular prism.**

Volume is given by the formula:

\begin{align*}\text{Volume}& = \text{length} \cdot \text{width} \cdot \text{height}\\ V & = l \cdot w \cdot h\end{align*}

You can calculate the **volume** of any **right rectangular prism** by multiplying the ________________________ of the solid, the ________________________ , and its _________________________ .

**Example 1**

*Find the volume of this prism:*

Use the formula for **volume** of a **right rectangular prism:**

\begin{align*}V &= l \cdot w \cdot h\\ V &= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \cdot \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \cdot \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ V &= 560\end{align*}

So the **volume** of this rectangular prism is 560 cubic units.

**Reading Check:**

1. *In your own words, what is volume?*

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

2. *True/False: An appropriate unit for the answer to a volume problem is cubic inches.*

3. *True/False: If 2 solids are congruent, then their volumes are the same.*

4. *True/False: Volume is calculated by taking the sum of the length, the width, and the height of a solid.*

## Volume of a Right Prism

Looking at the **volume** of **right prisms** with the *same* height and *different* bases, you can see a pattern. The computed area of each base is given below. The height of all 3 solids is the same, 10.

Putting the data for each solid into a table, we get:

Solid |
Height |
Area of base |
Volume |
---|---|---|---|

Rectangle | 10 | 300 | 3000 |

Trapezoid | 10 | 140 | 1400 |

Triangle | 10 | 170 | 1700 |

The relationship in each case is clear: when you multiply the height of the solid by the area of its base, you get the volume. This relationship can be proven to establish the following formula for any right prism.

The **volume** of a **right prism** is:

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

where \begin{align*}B\end{align*} is the **area** of the **base** of the 3-dimensional figure and \begin{align*}h\end{align*} is the prism’s **height** (also called **altitude**)

To find the **volume** of a **right prism**, you ______________________________ the area of its ________________________ by the __________________________ of the prism.

**Example 2**

*Find the volume of the prism with a triangular equilateral base and the dimensions shown in centimeters.*

To find the **volume**, first find the **area** of the **base**. In this diagram, the base is actually facing forwards instead of on the bottom. The **base** is an equilateral triangle as the directions say, so we use the area of a triangle formula:

\begin{align*}A = \frac{1}{2}\ bh\end{align*}

The **height** (or **altitude**) of the triangle is 10.38 cm. Each side of the triangle measures 12 cm. So the triangle has the following area:

\begin{align*}A &= \frac{1}{2}\ bh\\ &= \frac{1}{2} (12)(10.38)\\ &= 62.28 \ cm^2\end{align*}

Now use the formula for the volume of the prism, \begin{align*}V = Bh\end{align*}, where \begin{align*}B\end{align*} is the area of the **base** (the area of the triangle) and \begin{align*}h\end{align*} is the **height** of the prism.

Remember that the "height" of the prism is the distance between the bases, so in this case the height of the prism is 15 cm. Imagine that the prism is lying on its side.

\begin{align*}V& = Bh\\ &= (62.28)(15)\\ &= 934.2\end{align*}

Thus, the volume of the prism is \begin{align*}934.2 \ cm^3\end{align*} (or cubic centimeters).

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