Calculating area can be an unintuitive task for both students and teachers, whether it be parallelograms or circular area. The question: “How can I work out an area?” is one of the most common questions asked by students in the classroom – here is a comprehensive guide for regular and irregular areas!

What does my child need to know about calculating area?

The National Curriculum has very clear guidance about what students need to know about calculating area. By the end of KS2 (year 6) students should be able to:

-calculate and compare the area of rectangles (including squares), including using standard units, square centimetres (cm²) and square metres (m²), and estimate the area of irregular shapes

-find the area of rectilinear shapes by counting squares

-recognise when it is possible to use formulae for area and volume of shapes

calculate the area of parallelograms and triangles

-relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae (in words or symbols) to do this.

In addition to this, area is a common topic in both the 11+ for Grammar schools and the Common Entrance exam used for top independent schools. Both exams usually involve using formulas for areas of triangles, rectangles, squares, parallelograms as part of more complex reasoning problems (e.g. for irregular shapes – see the bottom of this guide).

So, now the question is simple: how can I work out an area?

How to work out the area of a circle

Calculating the area of a circle is easy although many students find it a strange formula. The circular area requires you to know only one length – the radius. This is the distance between the centre of the circle and the edge. The picture below shows the radius as the red line ‘R’. We are interested in the distance of R!

Once we have R all we need to do is to plug it into the famous formula which tells us how to work out the area of a circle:

A = π R²

This means that we square R (i.e. we do R x R) and then times that by pi (π) which roughly equals 3.14.

So, for example, if the radius is 5cm, the area is:

A = π R²

A = π x 5²

A = π x 25

A = 78.5 (to one decimal place!)

And really that’s all there is to it! Once you find R just plug it in!

How to work out the area of a rectangle

The general formula for calculating the area of a rectangle is simple!

Area = Length x Width

A = L x W

So, looking at the rectangle below the length is 10cm, and the width is 6cm.

A = L x W

A = 10 x 6

A = 60

So the area is 60cm²

How to work out the area of a square

What is a square? A square is the same as a rectangle except the length and the width are the same!

So, calculating the area of a square is simple. First we take the formula for the area of a rectangle:

Area = L x W

And  then we remember that the Length and Width are the same, so we can change it to this:

Area = L x L

A = L²

For example – the area of the square below, with side length of 10m is:

Area = L²

A = 10²

A = 100

How to work out the area of a triangle

A triangle is always half a rectangle! This little piece of information gives us all we need to know to work out the area – but let’s see why it’s true first.

Look at the blue triangle above. We can split that into two other triangles which are easier to work out the area! I’ve split it into the triangle in the red rectangle and the purple rectangle.

The triangle in the red rectangle makes up half the area of the red rectangle – because there is a straight line going from one corner to the other. So the area of that section is half the area of the rectangle

A = 4 x 1 ÷ 2

A = 2

The area of the triangle in the purple section is half the purple rectangle. So we can also work that out

by using the rectangle formula:

A = 4 x 6 ÷ 2

A = 12

Therefore the overall area is 12 + 2 = 14.

However – if we look at the original triangle, we realise that we don’t need to do all this fuss. We can just measure the base and the height and half that:

A = 4 x 7 ÷ 2

A = 14

So to summarise, the formula for the area of a triangle is:

Area = ½ x Base x Height

A = ½ x B x H

How to work out the area of a parallelogram

Parallelograms are a little bit more complicated as we need to make sure we are measuring the right distances before we do the calculation for the area!

So, before you read on – have a look at the parallelogram above and ask yourself which two are we interested in out of:

distance ‘a’                                          distance ‘b’                                          distance ‘h’

The key to getting this right is to think of the parallelogram as a rectangle that has been stretched! But the area stays the same.

So we only need to know the height (distance h) and the length (distance b).

And, in fact, the formula is the same as for a rectangle:

Area = base x height

A = b x h

How to calculate the area of an irregular shape

There are two ways to calculate the area of irregular shapes.

Firstly, you can divide the irregular shape into regular shapes and then calculate the area of these and add them up.

The second option is to treat the shape as a big regular shape, and then take away the extra irregular bits.

For example, with the shape below – you could treat it as a rectangle with a square on top.

Area (rectangle) = 3 x 5 = 19

Area (square) = 2 x 2 = 4

Overall area = 15 + 4 = 19

Or you could treat it as a big square with two holes in either corner.

Area (big square) = 5 x 5 = 25

Area (square hole top left) = 2 x 2 = 4

Area (rectangular hole top right = 1 x 2 = 2

Overall area = 25 – 4 – 2 = 19

So there are two different ways to find the area of this irregular shape: as long as you work logically and carefully and split the shape into regular shapes, you will have no problem!

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