## Areas of Plane Shapes

The areas of these shapes are shown by the shaded bit:

Area is a different type of measurement to say a measurement like length.  Length is a one-dimensional measurement, it measures how big a line is like this:

Area  is a two-dimensional  measurement.  Think of two length measurements, one horizontal and one vertical.  These two dimensions together measure out an area:

Now what happens to units when we measure areas.  Well, say we have an area measuring 1 metre by 1 metre:

To calculate an area like this, we need to multiply the two sides together.  So we get:

When you multiply metres by metres you get metres squared, also called square metres.  The ‘square’ is a reference both to the square shape of the area, and also to the fact that the ‘m’ is squared by raising it to the power 2.

Say you have a larger area like this:

You can split it up into smaller squares with each square having an area of 1 square metre:

The total area of this shape can be calculated by adding up each square area.  There are eight of those in total, so the total area is 8 m2.

### Converting between different units of area

You’ve got be really careful when you change from one type of unit to another when you’re doing areas.  It was easy with lengths – say you wanted to change from metres to centimetres.  You just needed to multiply by 100.  So 5 metres became 500 centimetres.  Well, with areas, you’ve got to remember that the units are squared – m2, cm2 etc…  So if you want to convert from say 1 m2 to square centimetres, look at this diagram:

1 metre has 100 centimetres in it, so along each side there are 100 little squares, each one representing 1 square centimetre.  How many square centimetres are there in a square metre?  Well if we have 100 square centimetres along one side, and 100 square centimetres along the other side, then we can get the total number of square centimetres by multiplying these together:

Yes, there are really 10,000 little squares in that diagram!  So when you convert between units measuring areas, do what you would do with lengths but do it twice.  So with lengths, to convert from cm to mm I’d multiply by 10.  With areas, to convert from cm2 to mm2 multiply by 10 twice, so by 100 in total.  Same applies in reverse too.  Say I wanted to convert from square metres to square kilometres.  Well, with lengths, I’d divide the number of metres by 1000 to get kilometres.  With areas I need to do this twice.  So to get from m2 to km2, I need to divide by 1000 twice, which is the same as dividing by 1,000,000 once.  This should give you an idea of how large a square kilometre really is – it’s huge.  There are 1,000,000 square metres in a square kilometre.  A normal suburban house and land has an area somewhere between 500 and 1000 square metres.  That means you could fit a whole lot of houses in one square kilometre!

### Imperial measurements of area

There are still a few non-metric measurements of area that are still in use.  In the housing industry, it’s very common for people to talk about acres or hectares.  People also sometimes talk about square inches, square feet, square yards and square miles.  Here are their equivalent measurements in metric units:

 Name Metric Equivalent 1 Acre 4,050 m2 1 Hectare 10,000 m2 1 Square Inch 6.45 cm2 1 Square Foot 929 cm2 1 Square Yard 0.836 m2 1 Square Mile 2.59 km2