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10.01 Units of length

Lesson

Introduction

When we measure the length of an object, we can express this length using distance units. The most common units of length we will encounter are:

  • millimetres (mm): a grain of sand has a length of around 1 to 3 mm.

  • centimetres (cm): the length of an ant is around 1 cm.
  • metres (m): the length of a desk is around 1 to 2 m.

  • kilometres (km): the Sydney Harbour Bridge is just over 1 km long.

We can use any of these units to measure any object, but some units are more convenient than others. For example, say we measured a football field to be 48\,737 mm wide. While this may be accurate, it isn't very helpful in giving us a visual idea of how wide the field really is. In order to convey this distance in an easier to understand unit we can apply a unit conversion.

Unit conversion

We can express the same length in different units, and to convert from one unit to another we will make use of the following relationships:

1 \text{ km} = 1000 \text{ m}

1 \text{ m} = 100 \text{ cm}

1 \text{ cm} = 10 \text{ mm}

Many rulers and tape measures show two units at once. It is common to label the cm markings with numbers, and to have mm markings in between each number to show that there are 10 mm in each centimetre.

A 10 centimetre ruler. Between each centimetre marking are ten 1 millimetre markings.

Examples

Example 1

Convert 6.52 centimetres to millimetres. Write your answer as a decimal.

Worked Solution
Create a strategy

Use the relationship 1 cm =10 mm.

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

We can express the same length in different units, and to convert from one unit to another we will make use of the following relationships:

1 \text{ km} = 1000 \text{ m}

1 \text{ m} = 100 \text{ cm}

1 \text{ cm} = 10 \text{ mm}

Conversion factors

Each unit conversion has a related conversion factor. In the example above we found that we can convert a length in cm to a length in m by dividing by 100.

Similarly, as there are 1000 m in 1 km, we can convert a length in km to a length in m by multiplying by 1000. A length in mm can be converted to a length in cm by dividing by 10. The conversions factors for common units are summarised in the image below.

Conversion factors between the units of length from millimetres to kilometres. Ask your teacher for more information.

Another way to think about using the conversion factor is to see that the number in the measurement changes in the opposite way to the unit. So when the unit gets bigger, the number gets smaller, and when the unit gets smaller, the number gets bigger.

Conversion between millimetres and centimetres and kilometres and metres. Ask your teacher for more information.

Examples

Example 2

Victoria is 1.14 m tall. Joanne is 156 cm tall.

a

Work out the height of Victoria in centimetres.

Worked Solution
Create a strategy

Use the relationship 1 m =100 cm.

Apply the idea
\displaystyle \text{Height}\displaystyle =\displaystyle 1.14 \times 100 \text{ cm}Multiply 1.14 by 100
\displaystyle =\displaystyle 114 \text{ cm}Evaluate
b

Who is taller?

Worked Solution
Create a strategy

Use the result found from part (a) and compare the value with Georgia's height.

Apply the idea

Since 114 \text{ cm } < 156 \text{ cm}, then this means that Joanne is taller than Victoria.

Idea summary

The conversions factors for common units of length are summarised below and can be used to perform conversions.

Conversion factors between the units of length from millimetres to kilometres. Ask your teacher for more information.

Converting units can also be useful when comparing two lengths that are given in two different units.

Mutliple conversions

We now know how to convert between adjacent units of length. What about converting from, say, millimetres to metres? Although we don't yet have a direct relationship for this conversion, we can make one from what we already know.

Using the fact that 1 \text{ m }= 100 \text{ cm}, and the fact that 1 \text{ cm }=10 \text{ mm}, we can combine these together to get the following relationship:

\displaystyle 1 \text{ m}\displaystyle =\displaystyle 100 \text{ cm}
\displaystyle =\displaystyle 100 \times 10\text{ mm}Use the fact that 1 \text{ cm }= 10\text{ mm}
\displaystyle =\displaystyle 1000 \text{ mm}Simplify the multiplication

Remember our football field? Let's try and make that distance make sense. So we have found that 1 \text{ m }=1000 \text{ mm}, and the conversion factor between millimetres and metres is 1000.

Examples

Example 3

Convert 512\,200 cm to km. Write your answer as a decimal.

Worked Solution
Create a strategy

Use the relationships 100\text{ cm}=1\text{ m} and 1000\text{ m}=1\text{ km}.

Apply the idea

To convert 512\,200 cm into km, divide this first by 100 to convert from centimetres into metres and then divide by 1000 to convert from metres into kilometres.

\displaystyle 512\,200 \text{ cm}\displaystyle =\displaystyle \dfrac{512\,200}{100} \text{ m}Divide 512\,200 by 100
\displaystyle =\displaystyle 5122 \text{ m}Evaluate
\displaystyle =\displaystyle \dfrac{5122}{1000}\text{ km}Divide 5122 by 1000
\displaystyle =\displaystyle 5.122 \text{ km}Evaluate
Idea summary

To perform multiple conversions we may need to use multiple operations as described in the image below:

Conversion factors between the units of length from millimetres to kilometres. Ask your teacher for more information.

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