# 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. The parallelogram can be considered as two congruent triangles. The area of the parallelogram is equal to the sum of areas of the two triangles. The parallelogram can be considered as a rectangle. The area of a parallelogram is therefore a product of its base and height.

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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