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10.08 Volume of rectangular prisms

Lesson

Introduction

The unit cube where each side is 1 unit and has a volume of 1 unit cubed.

The volume of a three dimensional shape is the amount of space that is contained within that shape.

A quantity of volume is represented in terms of the volume of a unit cube, which is a cube with side length 1 unit. By definition, a single unit cube has a volume of 1 cubic unit, written as 1\text{ unit}^3.

The image below shows a rectangular prism with length 5 units, width 3 units, and height 2 units. Notice that the length of each edge corresponds to the number of unit cubes that could be lined up side by side along that edge.

This image shows a rectangular prism broken down into cubes. Ask your teacher for more information.

We can find the number of unit cubes that could fit inside the rectangular prism by taking the product of the three side lengths. This gives 5\times3\times2=30, so there are 30 unit cubes in the prism, which means it has a volume of 30\text{ unit}^3.

Volume of a rectangular prism

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).

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 }\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.

A cube where three edges are labeled as side.

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.

A rectangular prism with length of 12 centimetres, width of 7 centimetres, and height of 5 centimetres.
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 hWrite the formula
\displaystyle =\displaystyle 12\times 7\times 5Subsitute the values
\displaystyle =\displaystyle 420 \text{ cm}^3Evaluate

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 hWrite the formula
\displaystyle =\displaystyle 24\times 12\times 19Subsitute the values
\displaystyle =\displaystyle 5472 \text{ cm}^3Evaluate
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.

An image with a box inside a larger size box that is twice the width, length and height.
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 5472Multiply the volume by 8
\displaystyle =\displaystyle 43\,776 \text{ cm}^3Evaluate
Idea summary

The volume of the rectangular prism is given by:

\displaystyle V=l\times w\times h
\bm{V}
is the volume of the rectangular prism
\bm{l}
is the length of the rectangular prism
\bm{w}
is the width of the rectangular prism
\bm{h}
is the height of the rectangular prism

The volume of the cube is given by:

\displaystyle V=s\times s\times s =s^3
\bm{V}
is the volume of the cube
\bm{s}
is the side length of the cube

Volume of composite shapes

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.

A composite shape built from rectangular prisms. Ask your teacher for more information.

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.

A composite shape with 5 specified measurements. Ask your teacher for more information.
Worked Solution
Create a strategy
Two smaller rectangular prisms from a composite shape. Ask your teacher for more information.

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+63Evaluate the multiplications
\displaystyle =\displaystyle 171 \text{ cm}^3Evaluate
Idea summary

The volume of a composite shape is the sum of the volume of each shape it is built from.

A composite shape built from rectangular prisms. Ask your teacher for more information.

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

Outcomes

MA4-14MG

uses formulas to calculate the volumes of prisms and cylinders, and converts between units of volume

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