Area of a Trapezium – GCSE Maths Revision Guide

The trapezium is one of the most commonly tested shapes at GCSE, and the good news is its area formula is given on every exam board's formula sheet. You still need to know how to use it correctly — identifying the parallel sides, choosing the perpendicular height, and applying the formula within compound shapes.

What Is a Trapezium?

A trapezium is a quadrilateral with exactly one pair of parallel sides. The two parallel sides are often labelled a and b, and the perpendicular distance between them is the height h.

Trapeziums appear in many real-world contexts — cross-sections of channels, sloping walls, and roof structures. In GCSE questions they frequently form part of a larger compound shape, so recognising them quickly is important.

The area formula works because a trapezium can be thought of as the average of the two parallel sides multiplied by the height — essentially treating it like a rectangle whose width is the mean of the top and bottom edges.

Key Formulas

A = ½(a + b) × h, where a and b are the parallel sides and h is the perpendicular height

For a compound shape containing a trapezium, find the trapezium area separately and add or subtract as needed

Step-by-Step Method

Identify the two parallel sides (a and b) — these are the sides that run in the same direction.
Find the perpendicular height (h) — the vertical distance between the two parallel sides, not a slant edge.
Substitute into the formula A = ½(a + b) × h and calculate.

Worked Example 1 — Foundation Level

Question: A trapezium has parallel sides of 6 cm and 10 cm and a perpendicular height of 4 cm. Find its area.

Working: A = ½(a + b) × h A = ½(6 + 10) × 4 A = ½ × 16 × 4 A = 8 × 4 A = 32

Answer: 32 cm²

Worked Example 2 — Higher Level

Question: A trapezium has an area of 60 cm². The parallel sides are 7 cm and 13 cm. Find the perpendicular height.

Working: A = ½(a + b) × h 60 = ½(7 + 13) × h 60 = ½ × 20 × h 60 = 10h h = 60 ÷ 10 h = 6

Answer: 6 cm

Worked Example 3 — Exam Style

Question: A compound shape is made from a rectangle and a trapezium. The rectangle measures 12 cm by 5 cm. The trapezium sits on top of the rectangle with parallel sides of 12 cm and 8 cm and a perpendicular height of 3 cm. Find the total area of the compound shape.

Working: Rectangle area = 12 × 5 = 60 cm² Trapezium area = ½(12 + 8) × 3 = ½ × 20 × 3 = 30 cm² Total area = 60 + 30 = 90

Answer: 90 cm²

Common Mistakes

Using a slant side instead of the perpendicular height. The height must be at right angles to both parallel sides, not along a sloping edge. Look for the right-angle symbol on the diagram.
Mixing up the parallel and non-parallel sides. Only the two parallel sides go into the formula — the other two sides are not used for the area calculation.
Forgetting to halve. The formula includes ½. Missing it will double your answer. Write the formula first to avoid this.
Not including units. Area is always measured in square units — cm², m², etc. Always state the correct unit in your final answer.

Exam Tips

The formula is on the formula sheet, but write it in your working — this earns a method mark even if you make an arithmetic slip.
If a trapezium appears inside a compound shape, deal with it separately and then combine areas at the end.
When the height is not labelled directly, look for a right-angle symbol or a dashed line — this indicates the perpendicular height.

Practice Questions

Q1 (Foundation): A trapezium has parallel sides of 9 cm and 15 cm and a perpendicular height of 6 cm. Find its area.

Answer: A = ½(9 + 15) × 6 = ½ × 24 × 6 = 72 cm².

Q2 (Foundation): A trapezium has parallel sides of 4.5 cm and 7.5 cm and a perpendicular height of 8 cm. Find its area.

Answer: A = ½(4.5 + 7.5) × 8 = ½ × 12 × 8 = 48 cm².

Q3 (Higher): A trapezium has an area of 84 cm². One parallel side is 10 cm and the perpendicular height is 7 cm. Find the length of the other parallel side.

Answer: 84 = ½(10 + b) × 7 → 84 = 3.5(10 + b) → 24 = 10 + b → b = 14 cm.

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Summary

A trapezium has exactly one pair of parallel sides.
The area formula is A = ½(a + b) × h, where a and b are the parallel sides and h is the perpendicular height.
The formula can be understood as the average of the two parallel sides multiplied by the height.
Always use the perpendicular height, not a slant edge.
The formula is provided on the exam formula sheet, but writing it in your working earns method marks.
Trapeziums often appear as part of compound shapes — calculate the trapezium area separately and combine.

Area of a Trapezium –