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Area of a Trapezium

FoundationHigherAQAEdexcelOCR

Master area of a trapezium for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers finding the area of a trapezium and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Area of a trapezium = ½(a + b) × h.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

A trapezium has exactly one pair of parallel sides, and its area is found with the formula \(A = \frac{1}{2}(a + b)h\), where \(a\) and \(b\) are the parallel sides and \(h\) is the perpendicular distance between them.

The formula works by averaging the two parallel sides and multiplying by the height, effectively turning the trapezium into a rectangle of the same area.

The height must be the perpendicular distance between the parallel sides, not the length of a slanted side. Diagrams often include a slanted length precisely to test whether you know the difference.

Revision notes

The formula

Add the parallel sides, halve the result, then multiply by the perpendicular height.

For parallel sides \(8\)cm and \(12\)cm with height \(5\)cm: \(\frac{1}{2}(8 + 12) \times 5 = 50\)cm².

Identifying the parts

The parallel sides are marked with arrows on the diagram. The height is the perpendicular gap between them.

Any slanted side given is usually a distractor, or is needed for the perimeter rather than the area.

Working backwards

If the area and one parallel side are known, substitute and solve for the missing value.

With area \(60\)cm², height \(6\)cm and one side \(7\)cm: \(60 = \frac{1}{2}(7 + b) \times 6\), giving \(b = 13\)cm.

Key points

  • A trapezium has one pair of parallel sides.
  • Area is \(\frac{1}{2}(a + b)h\).
  • \(a\) and \(b\) are the parallel sides.
  • \(h\) is the perpendicular height between them.
  • Arrows on the diagram mark the parallel sides.
  • A slanted side is not the height.

Worked examples

Example 1

Find the area of a trapezium with parallel sides \(6\)cm and \(10\)cm and height \(4\)cm.

Working

\[\frac{1}{2}(6 + 10)\]add the parallel sides and halve
\[8 \times 4\]multiply by the height
\[= 32\text{cm}^2\]state the area

Example 2

Find the area of a trapezium with parallel sides \(5\)cm and \(9\)cm and height \(6\)cm.

Working

\[\frac{1}{2}(5 + 9) = 7\]average the parallel sides
\[7 \times 6\]multiply by the height
\[= 42\text{cm}^2\]state the area

Example 3

A trapezium has area \(45\)cm², height \(5\)cm and one parallel side \(4\)cm. Find the other.

Working

\[45 = \frac{1}{2}(4 + b) \times 5\]substitute into the formula
\[18 = 4 + b\]multiply out and simplify
\[b = 14\text{cm}\]solve for the missing side

Common mistakes

  • Using a slanted side as the height.

    The height is the perpendicular distance between the parallel sides.

  • Forgetting to halve.

    The formula averages the parallel sides, so the ½ is essential.

  • Adding the wrong pair of sides.

    Only the two parallel sides are added, not all four.

  • Omitting squared units.

    Area answers need cm² or m².

Exam tips

  • Identify the parallel sides from the arrows before starting.
  • Write the formula out before substituting.
  • Ignore any slanted length unless the question asks for perimeter.
  • Include squared units in the answer.

Key terms

Trapezium
A quadrilateral with exactly one pair of parallel sides.
Parallel sides
The two sides marked with arrows, used as \(a\) and \(b\).
Perpendicular height
The distance between the parallel sides at right angles.
Average
The mean of the two parallel sides.

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Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.