The area of an irregular shape can sometimes be worked out or estimated by counting squares.

These shapes have an area of 2 square units. Part-squares can be joined together to make full squares when counting.

On other occasions estimation may be necessary. See problem 2 in the next section.

What this might look like in the classroom

Problem 1: Pick’s theorem

Draw some shapes on square dotty paper with one dot inside them. Make sure you join the dots with straight lines when you draw the shapes. Work out the area of each shape. Challenge pupils to find a rule connecting the area of the shape and the number of dots on the perimeter.

Now try shapes with 2 dots inside. Can you find a general rule connecting the area, the number of dots inside a shape and the number of dots on the perimeter?

Problem 2: How big is Cornwall?

Trace a map of Cornwall onto squared paper. Use the tracing to make an estimate of the area of Cornwall in square kilometres.

Answer:

Cornwall is approximately 3500 km^{2}.

Taking this mathematics further

Find out the formula for finding the area of some less common shapes. What does Heron’s Formula find the area of?

The process of integration finds the area under a curve bounded by the x-axis and by two vertical lines.

Methods of finding areas have practical uses where one needs to calculate how much carpet, paint or turf to buy for example, or how much of a chemical is needed to cover a certain area.

More sophisticated methods of estimating irregular areas, for example the area under a curve, are by using the Trapezium Rule or Simpson’s Rule.

Making connections

Area is defined as a measure of the amount of space occupied by a 2D shape or surface. It is measured in square units.

The area of an irregular shape can be found in two main ways. The area can be found by dividing the shape into rectangles (or counting squares). Alternatively it can be found by making an estimate. One way of doing this is to draw the shape on squared paper and to only count those squares that are more than half inside the shape.

Some pupils get confused with perimeter and area and fail to distinguish between them correctly. It is important that students are fluent in their understanding that perimeter is a measure of length and area is a measure in two dimensions.