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16.04 Metric units for area and volume

Lesson

Introduction

When measuring the length, area, or volume of an object, we want to choose the most appropriate unit. Sometimes the units we are given are not the easiest to deal with, they may be too large or too small and make calculations cumbersome.

A blue whale is 31\,456 \text{ mm} long, a one dollar coin has an area of 0.000\,491 square metres, and an apple has a volume of 0.000\,24 cubic metres. These are all correct measurements, but are they useful? Can you visualise these sizes in your mind?

Now, what if you were told a blue whale is about 32 \text{ m} long, a one dollar coin has an area of 491 \text{ mm}^2, which is a little bit less than 5 square centimetres, or that an apple has a volume of 240 cubic centimetres. These numbers are much more sensible, can make it much easier to visualise and also to make comparisons to other sizes. Sometimes, as in the case of the coin, two different units might both be considered appropriate.

Unit conversion

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

Example 1

Convert 6.52\text{ cm} to millimetres.

Worked Solution
Create a strategy

Use the conversion 1 \text{ cm}= 10 \text{ mm}.

Apply the idea

We need to multiply the centimetres by 10.

\displaystyle 6.52 \text{ cm}\displaystyle =\displaystyle 6.52 \times 10 \text{ mm}Multiply by 10
\displaystyle =\displaystyle 65.2 \text{ mm}Evaluate
Idea summary

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}

Area and volume conversions

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

Example 3

The rectangle below has side lengths given in centimetres.

Rectangle with 385 centimetres of width and 500 centimetres of length.
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
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 wUse the area of rectangle formula
\displaystyle =\displaystyle 5\times3.85Substitute l=5 and w=3.85
\displaystyle =\displaystyle 19.25 \text{ m}^2Evaluate

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 1000Multiply by 1000
\displaystyle =\displaystyle 9770 \text{ mm}^3Evaluate
Idea summary

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}

Volume conversions:

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

Hectares

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.

An image of two football fields side by side.

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.

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Examples

Example 5

Express 54\,800\text{ m}^2 in hectares.

Worked Solution
Create a strategy

Use the fact that 1\text{ ha}=10\,000\text{ m}^2, and divide by the conversion factor since we are converting to a larger unit.

Apply the idea
\displaystyle 54\,800\text{ m}^2\displaystyle =\displaystyle \dfrac{54\,800}{10\,000}\text{ ha}Divide by 10\,000
\displaystyle =\displaystyle 5.48\text{ ha}Evaluate
Idea summary

Hectare conversion:

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

A hectare is a unit used to describe area of any shape with 1 \text{ hectare}= 10\,000 \text{ m}^2.

Outcomes

MA4-13MG

uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area

MA4-14MG

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

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