# Area of a Trapezium Using Rectangles

To find the area of a trapezium students were previously provided with a formula.

Area of a trapezium = 1/2 (a + b)h

Even with this formula they would struggle to gain full marks when asked to find the area of a trapezium in an exam. For example, here is a question and examiner’s report by AQA.

ABCD is a trapezium.

Calculate the area of ABCD.

Examiner report

Most students used the formula but some confused addition and multiplication or omitted the factor of a half. Some arithmetic errors were seen, particularly from students working out 15 × 8 rather than 30 × 4. Some students simply evaluated 20 × 8.

The current specification does not provide students with the formula so a different approach is needed.   I teach this by drawing on their spatial reasoning and knowledge of rectangles.

## Beginning with a Parallelogram

We begin by finding the area of a parallelogram.  Using their whiteboards, I ask students to investigate another shape that would have the same area as a parallelogram.  The area of the shape they create must provide a simpler way of calculating the area of a parallelogram.

The diagrams below show the two most common arrangements.

After a short discussion the students agree it is easier to make a parallelogram into a rectangle to find its area.

## Area of a Trapezium

I now pose a trapezium as shown.

To find the area of a trapezium I ask the class to consider it as half of something else, whose area they are to find.   About half of the class created a hexagon as shown.

I asked the students to explain how finding the area of the hexagon is easier than the original trapezium.  Students agreed it was not easier.

After a few more minutes most students had formed the parallelogram or rectangle as shown.

With no further prompts from me I asked class to calculate the area of the original trapezium using their diagram.  About one-third found the area of the trapezium as 64 cm2 and two-thirds calculated 32 cm2.  I encouraged students to share their approaches and reconsider if needed.  Students who calculated 64 cm2 realised to half the 64 cm2. Click here to watch the video.

## Parallelograms and trapezia

To consolidate their learning students worked independently through the questions on the slide below in their books.  I asked the class to sketch each of the four diagrams in their book as the minimum for their working out.

## Extending the area of a trapezium

As learning progressed students used the area of a trapezium to calculate a missing length.

To begin with most of the class forgot to double the area of the trapezium.  I asked students to sketch the diagrams as they had previously.

By presenting their working out in this way all students correctly calculated the perpendicular height as 4 cm.

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

### Area of Compound Shapes

Read more about teaching the area of rectangles in this blog on exploring different ways to find the area of a compound shape.

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