WACE 11 Essential Mathematics - 2020 Edition 4.03 Units of area
Lesson

## Units of area

When we have a 2D shape, the amount of space it takes up is the area of that shape. This area is measured by comparing the shape to a square of fixed size. Thus the standard units for area are square units with familiar side lengths.

From smallest to largest the common units used for measuring area are:

• square millimetres (mm2)
• square centimetres (cm2)
• square meters (m2)
• square kilometres (km2)

Just as with lengths, while we can use any of these units to measure an area, it makes the most sense to use a unit that makes the measurement easy to visualise. To be able to visualise and estimate areas, it can be useful to identify common objects that are about the area of these units. For example:

• $1$1 mm2 is about the size of a piece of glitter.
• $1$1 cm2 is about the area of a fingernail.
• $1$1 m2 is roughly the area of floor below a shower, or that of a school desk.
• $1$1 km2 is the surface area of $800$800 Olympic swimming pools, or the area of several city blocks.

#### Practice questions

##### Question 1

It is most appropriate to use square metres (m2) to measure the area of:

A

Australia

B

A matchbox

C

An exercise book

D

A

Australia

B

A matchbox

C

An exercise book

D

##### Question 2

It is most appropriate to use square centimetres (cm2) to measure the area of:

A

A school playground

B

C

Japan

D

A

A school playground

B

C

Japan

D

Remember

Sometimes, it might occur that two different units of measurement are appropriate. In that case, you can choose either unit to use.

The area of some leaves, for example, could be measured with either mm2 or cm2.

## Estimating area

Sometimes we may only want an idea of the area of something, without needing an exact measurement. In such a case there are some techniques we can use to estimate area:

• Using a mental image of a common objects of size $1$1 mm2, $1$1 cm2, $1$1 m2 and/or $1$1 km2, try to picture how many it would take to cover the object.
• For shapes that are rectangular (or close to rectangular), we can estimate the length and width of the object and then multiply: $\text{Area}=\text{length}\times\text{width}$Area=length×width.
• If a picture of the item is available, you could draw a scale grid over it and estimate by counting the number of squares needed to cover the item.

#### Practice questions

##### Question 3

Estimate the area of the artwork above the sofa if the sofa is $1.9$1.9m in length. 1. $1.7$1.7 m2

A

$0.01$0.01 m2

B

$5$5 m2

C

$0.5$0.5 m2

D

$1.7$1.7 m2

A

$0.01$0.01 m2

B

$5$5 m2

C

$0.5$0.5 m2

D

##### Question 4

Estimate the area of the curved shape below if each square on the grid has an area of $3$3 mm2. 1. $111$111 mm2

A

$72$72 mm2

B

$26$26 mm2

C

$48$48 mm2

D

$111$111 mm2

A

$72$72 mm2

B

$26$26 mm2

C

$48$48 mm2

D

##### Question 5

Which of these objects do you think would have an area of about $1$1m2?

1. goal area on a football field

A

lid of a shoe box

B

coffee table

C

baking tray

D

goal area on a football field

A

lid of a shoe box

B

coffee table

C

baking tray

D

## Convert units of area

Converting units of area is like converting units of length, but this time we want to know how many smaller square units fit into the larger square unit. The following applet can help visualise and understand this type of conversion. Use the slider to have a look at some different conversions:

 Created with Geogebra

To convert units of area, such as from square millimetres (mm2) to square centimetres (cm2), it may help to think of how to convert units of length first. The following video explains how to convert units of area.

The conversion factors between the different area units are: A simple way to remember these factors is to remember the conversion factors for lengths and square the factor for conversion of units of area. If we know that the conversion from cm to m is divided by $100$100, then to convert from cm2 to m2 we divide by $100$1002, or divide by $10000$10000.

Remember!

When converting between units of area:

• Multiply if converting to a smaller unit - more smaller squares will be needed to cover the same area
• Divide if converting to a larger unit - less larger squares will be needed to cover the same area
• Multiply or divide by the conversion factor for lengths squared

Length conversions (learn this!): Area conversions (square the numbers above): #### Worked example

##### Example 1

Convert $29800$29800 cm2 to m2.

Think: We are converting from cm2 to m2, since we are converting to a larger unit we need to divide. The conversion factor between centimetres and metres is $100$100. So we need to divide by $100$100 squared.

Do:

 $29800$29800 cm2 $=$= $29800\div100\div100$29800÷​100÷​100 m2 $=$= $2.98$2.98 m2

#### Practice questions

##### Question 6

Select the option that shows $3$3 m2 converted into cm2.

1. $3000$3000 cm2

A

$300000$300000 cm2

B

$30000$30000 cm2

C

$300$300 cm2

D

$3000$3000 cm2

A

$300000$300000 cm2

B

$30000$30000 cm2

C

$300$300 cm2

D

##### Question 7

Convert $0.56$0.56km2 to m2.

## Other units of area

Just as with length there are other units of area that were traditionally used and some are still in use today in a few countries or used for particular applications.

For large areas of land such as farms or national parks the unit hectares (ha) is sometimes used. This is actually a metric unit, with $1$1 hectare being equivalent to $10000$10000 m2. An imperial unit used for large areas of land is acres (ac), which is $\frac{1}{640}$1640 square miles or approximately $0.4$0.4 hectares. The imperial units we encountered in lengths, which were inches, feet, and miles, can all be used to form units for area - square inches (in2), square feet (ft2), and square miles (mi2). If given the conversion factor, we can convert between units that we are less familiar with, such as these imperial units.

#### Worked example

##### Example 3

A block of land is listed as $2.5$2.5 hectares.

(a) If $1$1 hectare is equivalent to $10000$10000 m2, how many square metres is the land?

Think: When converting unfamiliar units it is a good idea to set up the given equivalence as an equation and then use operations, keeping the equation balanced, to find the required amount.

Do:

 $1$1 ha $=$= $10000$10000 m2 Write the conversion factor in the form of an equation. $2.5$2.5 ha $=$= $2.5\times10000$2.5×10000 m2 Multiply both sides by $2.5$2.5. $2.5$2.5 ha $=$= $25000$25000 m2

(b) If $1$1 acre is approximately $4000$4000 m2, how many acres is the property?

Think: Again let's set this up as an equation and work step by step to find how many acres $25000$25000 m2 is.

Do:

 $4000$4000 m2 $=$= $1$1 acre Write the conversion factor in the form of an equation. $\frac{4000}{4000}$40004000​ m2 $=$= $\frac{1}{4000}$14000​ acres Find the equivalent of $1$1 m2 by dividing both sides by $4000$4000. $1$1 m2 $=$= $0.00025$0.00025 acres $25000$25000 m2 $=$= $25000\times0.00025$25000×0.00025 acres Multiply both sides by $25000$25000. $\therefore25000$∴25000 m2 $=$= $6.25$6.25 acres

#### Practice questions

##### Question 8

Express $54800$54800m2 in hectares.

##### Question 9

A property covers an area of $4.5$4.5 hectares. Given that $1$1 hectare is $10000$10000 m2, determine the following:

1. The area of the property in m2.

2. The area of the property in acres, given that $1$1 acre is approximately $4000$4000 m2.

##### Question 10

American football is played on a field with an area of $12000$12000 square yards.

1. Given that $1$1 square yard is approximately $0.836$0.836 m2, determine the area of an American football field in m2.

2. The MCG oval has an area of $17700$17700 m2. How much smaller (in m2) than the MCG oval is an American football field?

### Outcomes

#### 1.3.5

choose and use appropriate metric units of area, their abbreviations and conversions between them

#### 1.3.6

estimate the areas of different shapes

#### 1.3.7

convert between metric units of area and other area units