11.04 Volume and units

When finding the volume of a solid using its dimensions, we need to first choose a common unit of measurement that all the dimensions will be in terms of. Once all the dimensions have been converted to this common unit, we can then calculate the volume as per usual.

Examples

When converting units of volume, we can think of it as converting the units of each dimension that multiply to make up that volume. Since volume is calculated using three dimensions, any conversion between units will need to be applied three times.

Consider a cube with a side length of 1 cm.

Since there are 10 mm in each 1 cm we know that ten cubes of side length 1 mm will fit along the length, base and height of the cube. In other words, one cubic centimetre has the same volume as 10 \times 10 \times 10 cubic millimetres.

1\text{ cm}^{3} = 1000 \text{ mm}^{3}

Now consider a cube with a side length of 1 m. Since 1 m is equal to 100 cm, we can fit a hundred cubic centimetres along each dimension of the cubic metre. So one cubic metre has the same volume as 100\times 100 \times 100 cubic centimetres.

1\text{ m}^{3} = 1\,000\,000 \text{ cm}^{3}

Now what if we combined these two conversion equations?

If there are 1000 cubic millimetres in a cubic centimetre, and there are 1\,000\,000 cubic centimetres in a cubic metre, then we get:

1\text{ m}^{3} = 1\,000\,000\, 000 \text{ mm}^{3}

Volume conversions: \begin{aligned} 1\text{ cm}^{3} &= 1000\text{ mm}^{3}\\1\text{ m}^{3} &= 1\,000\,000\text{ cm}^{3}\\1\text{ m}^{3} &= 1\,000\,000\,000\text{ mm}^{3}\end{aligned}

To convert from a larger unit to a smaller unit, we multiply by the conversion factor. To convert from a smaller unit to a larger unit, we divide by the conversion factor.

Examples

Example 2

Complete the working below to convert 4\text{ m}^{3} to a volume in \text{ mm}^{3}.

\displaystyle 4\text{ m}^{3}	\displaystyle =	\displaystyle 4 \times 1\,000\,000 \text{ cm}^{3}	Multiply 4 by 1\,000\,000 \text{ cm}^{3}
	\displaystyle =	\displaystyle ⬚\text{ cm}^{3}	Evaluate
	\displaystyle =	\displaystyle ⬚ \times 1000 \text{ mm}^{3}	Multiply the number of \text{cm}^{3} by 1000\text{ mm}^{3}
	\displaystyle =	\displaystyle ⬚\text{ mm}^{3}	Evaluate

Worked Solution

Create a strategy

We need first to convert \text{ m}^{3} to \text{ cm}^{3}, and then convert \text{ cm}^{3} to \text{ mm}^{3}.

Apply the idea

\displaystyle 4\text{ m}^{3}	\displaystyle =	\displaystyle 4 \times 1\,000\,000 \text{ cm}^{3}	Multiply 4 by 1\,000\,000 \text{ cm}^{3}
	\displaystyle =	\displaystyle 4\,000\,000\text{ cm}^{3}	Evaluate
	\displaystyle =	\displaystyle 4\,000\,000 \times 1000 \text{ mm}^{3}	Multiply the number of \text{cm}^{3} by 1000\text{ mm}^{3}
	\displaystyle =	\displaystyle 4\,000\,000\,000\text{ mm}^{3}	Evaluate

Watch question walkthrough

Example 3

The outline of a trapezium-shaped block of land is pictured below.

a

Find the area of the block of land in square metres.

Worked Solution

Create a strategy

We can use the area of trapezium formula: A=\dfrac{h}{2}\,(a+b).

Apply the idea

\displaystyle \text{Area}	\displaystyle =	\displaystyle \dfrac{6}{2}\,( 8+16)	Substitute h,\,b,\, and a
	\displaystyle =	\displaystyle 72\text{ m}^{2}	Evaluate

Watch question walkthrough

b

During a heavy storm, 41 mm of rain fell over the block of land. What volume of water landed on the property in litres?

Worked Solution

Create a strategy

Convert \text{mm} to \text{ m} then \text{ m}^{3} to litres.

Apply the idea

Use the fact that 1\text{ mm} = \dfrac{1}{1000}\text{ m}, and 1\text{ m}^{3} = 1000\text{ L} to get:

\displaystyle \text{Volume}	\displaystyle =	\displaystyle 72\times \dfrac{41}{1000}\text{ m}	Substitute A and h in metres
	\displaystyle =	\displaystyle \dfrac{2952}{1000} \text{ m}^3	Evaluate
	\displaystyle =	\displaystyle \dfrac{2952}{1000} \times 1000 \text{ L}	Convert to litres
	\displaystyle =	\displaystyle 2952 \text{ L}	Evaluate

Watch question walkthrough

Similar to the way that we can convert between units of volume, we can also scale volume by applying the scale factor three times. To scale area we apply the scale factor two times.

Examples

Example 4

Jeremy has an old box with dimensions of 10\text{ cm}, 12\text{ cm} and 4\text{ cm}, which has a volume of 480\text{ cm}^{3}. If Jeremy makes a new box with dimensions double that of his old box, what will its volume be?

Worked Solution

Create a strategy

Double the dimensions of the old box to get the dimensions of the new box and then multiply them together to find the volume.

Apply the idea

The dimensions of the new box are: 20\text{ cm},\,24\text{ cm} and 8\text{ cm}.

\displaystyle \text{Volume}	\displaystyle =	\displaystyle 20\times 24\times 8	Multiply the dimensions
	\displaystyle =	\displaystyle 3840\text{ cm}^{3}	Evaluate

Reflect and check

Notice that the volume of the new box is equal to 8 times the volume of the old box. This is because each dimension of the new box is double a dimension of the old box. This was the same as multiplying by 2, three times. Since 2\times 2\times 2 = 8, the volume of the new box must have been 8 times greater than the volume of the old box.