GCSE MATHS • HIGHER TIER
Area of Trapeziums Made Simple
Learn how to use variables and algebraic reasoning to prove mathematical statements are always true.
Clear method
Exam practice
Worked proofs
Proof in Four Moves
1
Identify
Find the two parallel sides. These are the lengths represented by a and b in the formula.
2
Find the Height
Use the perpendicular distance between the parallel sides, not a sloping side.
3
Substitute
Place a, b and h into A = ½(a + b)h, keeping the brackets around a + b.
4
Calculate and Label
Add the parallel sides, multiply by the height, divide by 2 and give the answer in square units.
What You’ll Learn
Recognise a Trapezium
Identify a quadrilateral with one pair of parallel sides and locate the measurements needed for its area.
Use the Area Formula
Apply A = ½(a + b)h accurately when both parallel sides and the perpendicular height are known.
Choose the Correct Height
Distinguish the perpendicular height from sloping sides that are not used in the area formula.
Find Missing Measurements
Rearrange the formula to calculate an unknown height or an unknown parallel side.
Check Units and Reasonableness
Use square units for area and test whether the final value is sensible for the given dimensions.
Key Representations
Area Formula
A = ½(a + b)h
Find the Height
h = 2A ÷ (a + b)
Find a Missing Parallel Side
b = 2A ÷ h − a
Area Units
cm², m², mm²
Worked Proof: Consecutive Numbers
1. Write the Formula
A = ½(a + b)h
2. Substitute the Known Values
120 = ½(12 + b) × 8
3. Simplify
120 = 4(12 + b)
4. Divide by 4
30 = 12 + b
5. Solve for b
b = 18 cm
More Worked Examples
1
Parallel Sides 8 cm and 12 cm
Formula: A = ½(a + b)h
Calculate:
A = ½ × 20 × 5
A = 50 cm²
2
Parallel Sides 10 cm and 16 cm
Formula: A = ½(a + b)h
Calculate:
A = ½ × 26 × 7
A = 91 cm²
3
Parallel Sides 14 cm and 20 cm
Formula: A = ½(a + b)h
Calculate:
A = ½ × 34 × 6
A = 102 cm²
4
Parallel Sides 18 cm and 24 cm
Formula: A = ½(a + b)h
Calculate:
A = ½ × 42 × 10
A = 210 cm²
- Common GCSE Mistakes
- Using a non-parallel side
- Using a sloping side as h
- Forgetting the half
- Losing the brackets
- Using length units
- Rearranging only part of the formula
- Exam Tips
- Mark the two parallel sides on the diagram before writing any formula.
- Draw or identify a right-angle marker to confirm the perpendicular height.
- Write A = ½(a + b)h before substituting so your method is clear.
- Keep units attached to the measurements and use square units for the area.
- Substitute your answer back into the original formula when finding a missing length.
Skills Used in Algebraic Proof
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Area of Trapeziums Tutorial
Step-by-step walkthroughs of Area of Trapeziums with clear methods and exam tips.
Frequently Asked Questions
A trapezium is a quadrilateral with one pair of parallel sides. In area questions, those parallel sides are normally labelled a and b. The other two sides may be sloping, vertical or different lengths.
Use A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height. Add the parallel sides first, multiply by the height and then divide by 2.
The height is the shortest perpendicular distance between the parallel sides. Look for a right-angle marker. A sloping side should not be used unless it meets the parallel sides at 90 degrees.
Substitute the known values into A = ½(a + b)h and solve the resulting equation. You may use h = 2A ÷ (a + b) for the height or b = 2A ÷ h − a for a missing parallel side.
