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 three dimensional shape 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 }\times \text{width }\times \text {height,\quad}\text{or}\\ V&=l\times w\times h \end{aligned}

A cube where three edges are labeled as side.

A cube can be thought of as a special type of rectangular prism, one that has all sides equal in length.

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.

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\times w\times h	Use the volume formula
	\displaystyle =	\displaystyle 14\times6\times4	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\times w\times h	Use the volume formula
	\displaystyle =	\displaystyle 3 \times \dfrac{7}{2} \times 6	Substitute l=3, w=\dfrac{7}{2}, and h=6
	\displaystyle =	\displaystyle \dfrac{3}{1} \times \dfrac{7}{2} \times \dfrac{6}{1}	Rewrite the whole numbers as fractions
	\displaystyle =	\displaystyle \dfrac{126}{2}	Multiply
	\displaystyle =	\displaystyle 63 \text { cm}^3	Simplify

Idea summary

The volume of a rectangular prism is given by:

\displaystyle V=l\times w\times h

\bm{V}

is the volume

\bm{l}

is the length

\bm{w}

is the width

\bm{h}

is the height

The volume of a cube is given by:

\displaystyle V=s\times s\times s=s^3

\bm{s}

is the side length of a cube

Units

The unit cube where each side is 1 unit and has a volume of 1 unit to the power of 3.

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.

We use special units to describe volume, based on the notion of cubic units described above. Because the units for length include millimeters, centimeters, meters and kilometers we end up with the following units for volume: \text { mm}^3, \text { cm}^3, \text { m}^3, etc.

Before we start a question, it is important to check that all of the sides are in the same unit. If they aren't, then we should convert them to the same unit.

Examples

Example 3

A box is 1 meter long, 20 centimeters high and 30 centimeters wide.

Determine the volume of the box in cubic centimeters.

Worked Solution

Create a strategy

Convert the length to centimeters then use the volume of a rectangular prism formula.

Apply the idea

\displaystyle l	\displaystyle =	\displaystyle ⬚ \text{ cm}	Consider how many centimeters are in 1 meter
\displaystyle l	\displaystyle =	\displaystyle 100 \text{ cm}	Convert length
\displaystyle V	\displaystyle =	\displaystyle l\times w\times h	Use the volume formula
	\displaystyle =	\displaystyle 100\times30\times20	Substitute l=100, w=30, and h=20
	\displaystyle =	\displaystyle 60\,000\text{ cm}^3	Evaluate

Idea summary

Before we start a question, it is important to check that all of the sides are in the same unit. If they aren't, then we should convert them to the same unit.

Units of Volume:

\text{cubic millimeters} = \text{mm}^3
\text{cubic centimeters} = \text{cm}^3
\text{cubic meters} = \text{m}^3

Outcomes

6.G.A.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas v = lwh and = bhv to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

7.07 Volume of rectangular prisms

Ideas

Volume of a rectangular prism

Examples

Example 1

Example 2

Units

Examples

Example 3

Outcomes

6.G.A.2

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