Measurement

New Zealand

Level 6 - NCEA Level 1

Lesson

We use special units to describe area, based on the notion of square units described ifarean this previous chapter. Because the units for length include millimetres, centimetres, metres and kilometres we end up with the following units for area.

**square millimetres = mm ^{2}**

(picture a square with side lengths of $1$1 mm each - pretty small this one!)

**square centimetres = cm ^{2}**

(picture a square with side lengths of $1$1 cm each - about the size of a fingernail)

**square metres = m ^{2}**

(picture a square with side lengths of $1$1 m each - what do you know that is about this big?)

**square kilometres = km ^{2} **

(picture a square with a side length of $1$1km - I wonder how many of these your town or city's land area measures?)

When converting units of area we need to work out how many smaller square units fit into the larger square unit. This applet can help us to visualise and understand this conversion. Have a look at the different conversions shown here by sliding the slider.

1 square kilometre = 1 km^{2}

Converting into square metres: $1000\times1000$1000×1000 m^{2} = $1000000$1000000m^{2}

What if we want to turn $1$1 km^{2} into cm^{2} (square centimetres)?

First we would have to work out how many centimetres are in a length of one kilometre:

$100$100 (centimetres in each metre) × $1000$1000 (metres in each kilometre) =$100000$100000.

Then we multiply the side lengths of the square kilometre: $100000\times100000=10000000000$100000×100000=10000000000cm^{2}. WOAH this is huge! This is exactly why we use different units of area. Imagine having to describe the size of your classroom in square centimetres.

1 square metre = 1 m^{2}

Converting into square centimetres: $100\times100=10000$100×100=10000 cm^{2}

Converting into square millimetres: $1000\times1000=1000000$1000×1000=1000000 mm^{2}

1 square centimetre = 1 cm^{2}

Converting into square millimetres: $10\times10=100$10×10=100 mm^{2}

When measuring land there are two other common units of area.

**hectare = ha**

$1$1 ha = $10000$10000m^{2}

(imagine a square $100$100 m x$100$100 m - this is the size of a hectare)

An acre is a very common unit for measuring blocks of land here in Australia, UK and indeed a lot of countries around the world. But $1$1 acre has a very bizarre conversion.

$1$1 acre =$4046.85642$4046.85642 m^{2}. Quite a bizarre number really, especially when all the numbers we have used so far in our conversions are multiples of $10$10 and nice to work with.

In fact an acre is an imperial unit, where $1$1 acre = $4840$4840 square yards. Whilst still not a multiple of $10$10 (like the units in our metric system are), it is still a lot nicer than $4046.85642$4046.85642 m^{2}.

Fortunately for us as budding mathematicians it is not common to convert between acres and metric measurements.

Express $10.4$10.4ha in m^{2}.

Convert $34000$34000cm^{2} to m^{2}.

A rectangular farm has an area of $12$12 ha and a length of $600$600 m. What is the width of the farm in metres?