16.04 Metric units for area and volume

Much like we can express the same length in different units, we can convert from one unit of area to another, and similarly from one unit of volume to another we will make use of the following relationships. Let's first remind ourselves of the length conversions, as we will use these to work out the conversions for area and volume.

Here are some of the length conversions:

\begin{array}{c} 1 \text{ km} &=& 1000 \text{ m} &=& 100\,000 \text{ cm}\\ 1 \text{ m} &=& 100 \text{ cm} &=& 10\,000 \text{ mm} \\ 1 \text{ cm} &=& 10 \text{ mm} &=& 0.01 \text{ m} \end{array}

Examples

Here are some of the area conversions:

\begin{aligned} 1 \text{ km}^2 &= 1\, 000\, 000 \text{ m}^2 \\ 1 \text{ m}^2 &= 10\, 000 \text{ cm}^2 \\ 1 \text{ cm}^2 &= 100 \text{ mm}^2 \end{aligned}

Here are some of the volume conversions:

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

Examples

Example 2

Convert 6 \text{ km}^2 into \text{m}^2.

Worked Solution

Create a strategy

Use the conversion 1 \text{ km}^2= 1\,000\,000 \text{ m}^2.

Apply the idea

\displaystyle 6 \text{ km}^2	\displaystyle =	\displaystyle 6 \times 1\,000\,000\text{ m}^2	Multiply by 1\,000\,000
	\displaystyle =	\displaystyle 6\,000\,000 \text{ m}^2	Evaluate

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Example 3

The rectangle below has side lengths given in centimetres.

a

Convert the dimensions of the rectangle into metres.

Worked Solution

Create a strategy

Use the conversion 1 \text{ m}= 100 \text{ cm}.

Apply the idea

\displaystyle 500 \text{ cm}	\displaystyle =	\displaystyle \dfrac{500}{100}	Divide by 100
	\displaystyle =	\displaystyle 5 \text{ m}	Evaluate
\displaystyle 385 \text{ cm}	\displaystyle =	\displaystyle \dfrac{385}{100}	Divide by 100
	\displaystyle =	\displaystyle 3.85 \text{ m}	Evaluate

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b

Find the area of the rectangle in square metres.

Worked Solution

Create a strategy

Use the area of a rectangle formula with the new dimensions.

Apply the idea

\displaystyle \text{Area}	\displaystyle =	\displaystyle l \times w	Use the area of rectangle formula
	\displaystyle =	\displaystyle 5\times3.85	Substitute l=5 and w=3.85
	\displaystyle =	\displaystyle 19.25 \text{ m}^2	Evaluate

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Example 4

Convert 9.77 \text{ cm}^3 into \text{mm}^3.

Worked Solution

Create a strategy

Use the conversion 1 \text{ cm}^3= 1000 \text{ mm}^3.

Apply the idea

\displaystyle 9.77 \text{ cm}^3	\displaystyle =	\displaystyle 9.77 \times 1000	Multiply by 1000
	\displaystyle =	\displaystyle 9770 \text{ mm}^3	Evaluate

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So far we have looked at units of volume and area that are based of our metric units of length, millimetres, centimetres, metres and kilometres.

But there is one special unit of area that does not quite fit the pattern of the others. It is a very useful measurement that helps fill the gap between a square metre (roughly the size of a desk) and a square kilometre (the size of several city blocks).

That's where the hectare comes in.

A hectare is square 100 \text{ m} by 100 \text{ m} - it helps to think of it as two football fields side by side.

Here is the hectare conversion:

1 \text{ hectare}= 100 \text{ metres} \times 100 \text{ metres}= 10\,000 \text{ square metres}

Hectares are useful for describing area for things like sporting grounds, farms and parks. To be able to remember the size of a hectare, it helps to see how it is derived, and how it compares to the square metre and square kilometre.

Examples