Create a base by moving the length and width sliders, and keeping the height slider at zero. What is the area of the base?
Slowly drag the height slider. What happens to the volume of the rectangular prism?
How does the volume of the prism change each time you make its height 1 unit bigger, and what does that have to do with the area of the base?
How can we find the volume of a rectangular prism if we already know how to calculate the area of its base using the area formula?

A rectangular prism with length of 5 units, width of 4 units and height of 2 units.

We can find the volume of a rectangular prism by first finding the area of the base and multiplying that by the height of the prism.

Notice that the figure has a rectangular base with side lengths of 5 units and 4 units. To find the area of the rectangular base, we multiply the length and width and get 20 square units.

Next, notice the prism is 2 layers high, so it has a height of 2 units. Now, we can multiply the area of the base by the height to calculate the volume.

\text{Volume } = \text{area of base} \cdot \text{height}

\text{Volume } = 20 \cdot 2

\text{Volume }= 40

In the same way that the area of a two dimensional shape is related to the product of two perpendicular lengths, the length and width, the volume of a rectangular prism is related to the product of three perpendicular lengths, the length, width, and height. Notice that each of the three lengths is perpendicular to the other two.

A rectangular prism where each side is labeled as length, width, and height.

The volume of a rectangular prism is given by

\begin{aligned} \text{Volume }&=\text{length }\cdot \text{width }\cdot \text {height,\quad}\text{or}\\ V&=l\cdot w\cdot h \end{aligned}

Examples

Example 1

Find the volume of the rectangular prism shown.

Rectangular prism with length of 14 centimeters, height of 4 centimeters, and width of 6 centimeters.

Worked Solution

Create a strategy

Use the volume of a rectangular prism formula.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle l\cdot w\cdot h	Use the volume formula
	\displaystyle =	\displaystyle 14\cdot6\cdot4	Substitute l=14, w=6, and h=4
	\displaystyle =	\displaystyle 336\text{ cm}^3	Evaluate

Example 2

Find the volume of the rectangular prism shown.

A rectangular prism with a length of 3 centimeters, width of 7/2 centimeters and height of 6 centimeters.

Worked Solution

Create a strategy

Even though we have a fractional edge length, we can use the same formula.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle l\cdot w\cdot h	Use the volume formula
	\displaystyle =	\displaystyle 3 \cdot \dfrac{7}{2} \cdot 6	Substitute l=3, w=\dfrac{7}{2}, and h=6
	\displaystyle =	\displaystyle \dfrac{3}{1} \cdot \dfrac{7}{2} \cdot \dfrac{6}{1}	Rewrite the whole numbers as fractions
	\displaystyle =	\displaystyle \dfrac{126}{2}	Multiply
	\displaystyle =	\displaystyle 63 \text { cm}^3	Simplify

Idea summary

A rectangular prism is a 3D shape with six rectangular faces.

rectangular prism labeled with 'Length','Height' and 'Width'.

The volume of a rectangular prism is given by:

\displaystyle V=l\cdot w\cdot h

\bm{V}

is the volume

\bm{l}

is the length

\bm{w}

is the width

\bm{h}

is the height

7.03 Review: Volume of rectangular prisms

Ideas

Volume of rectangular prisms

Exploration

Examples

Example 1

Example 2

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