10.08 Volume of rectangular prisms

In the same way that the area of a two dimensional shape is related to the product of two perpendicular lengths, the volume of a three dimensional shape is related to the product of three mutually perpendicular lengths (each of the three lengths is perpendicular to the other two).

The volume of a rectangular prism is given by

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

A cube can be thought of as a special type of rectangular prism, one that has all sides equal in length. The formula for the volume of a cube is similar to the formula for the area of a square.

The volume of a cube is given by

\begin{aligned} \text{Volume }&=\text{side }\times \text{side }\times \text{side,\quad}\text{or}\\ V&=s\times s\times s=s^3 \end{aligned}

Examples

Example 1

Find the volume of the rectangular prism shown.

Worked Solution

Create a strategy

Use the formula for the volume of a rectangular prism.

Apply the idea

Based on the diagram above, we are given l=12, \, w=7 and h=5.

\displaystyle V	\displaystyle =	\displaystyle l\times w\times h	Write the formula
	\displaystyle =	\displaystyle 12\times 7\times 5	Subsitute the values
	\displaystyle =	\displaystyle 420 \text{ cm}^3	Evaluate

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Example 2

A box of cereal is in the shape of a rectangular prism. It measures 24 cm by 12 cm by 19 cm.

a

What is the volume of the box of cereal?

Worked Solution

Create a strategy

Use the formula for the volume of a rectangular prism.

Apply the idea

We are given l=24,\, w=12 and h=19.

\displaystyle V	\displaystyle =	\displaystyle l\times w\times h	Write the formula
	\displaystyle =	\displaystyle 24\times 12\times 19	Subsitute the values
	\displaystyle =	\displaystyle 5472 \text{ cm}^3	Evaluate

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b

The company that makes these cereal boxes also makes a jumbo size box, which is twice as long, twice as wide, and twice as tall as the regular size boxes. What is the volume of the jumbo box of cereal?

Worked Solution

Create a strategy

Use this image which shows how one regular size box would fit into a jumbo size box.

Apply the idea

Since the length, width, and height have been doubled we need to multiply the volume by \\ 2\times 2\times 2=8.

\displaystyle V	\displaystyle =	\displaystyle 8 \times 5472	Multiply the volume by 8
	\displaystyle =	\displaystyle 43\,776 \text{ cm}^3	Evaluate

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Once we are familiar with finding the volume of rectangular prisms and cubes, the same idea can be used to determine the volume of more complicated shapes. If we think of a composite shape as being built out of a number of smaller, simpler shapes, then the volume of the composite shape is the sum of the volume of each shape it is built from.

This complicated shape is built from simple rectangular prisms.

Depending on the configuration of the composite shape, it may be useful to think of building it from a large rectangular prism that then has smaller volumes taken away.

Examples

Example 3

Find the volume of the composite solid shown by breaking it up into rectangular prisms.

Worked Solution

Create a strategy

Break up the composite shape into two smaller rectangular prisms, as shown.

The volume of the entire shape is found by adding together the volume of the two smaller rectangular prisms.

Apply the idea

\displaystyle V_{\text{composite}}	\displaystyle =	\displaystyle V_{\text{top prism}}+V_{\text{bottom prism}}	Add the two smaller volumes
	\displaystyle =	\displaystyle \left(12\times 3\times 3\right)+\left(3\times 3\times 7\right)	Substitute the values
	\displaystyle =	\displaystyle 108+63	Evaluate the multiplications
	\displaystyle =	\displaystyle 171 \text{ cm}^3	Evaluate

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