Trapezium areas and missing dimensions
Average the parallel sides and multiply by their perpendicular separation.
Section Paper 2 · Extended · Non-calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Calculate a mean of two numbers.
- Rearrange a simple formula.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the average of 6 and 10?
What you will learn
- Recognise the two parallel sides of a trapezium.
- Find area and recover a missing height or parallel side.
Average the parallel widths
A trapezium with parallel sides a and b and perpendicular separation h has area (a + b)h/2. Two matching trapezia can form a parallelogram of base a + b and height h, so one occupies half that area. A and b must be the parallel pair; the two sloping sides do not belong in this formula.
With parallel sides 6 cm and 10 cm and height 5 cm, the average width is 8 cm, giving area 40 cm². The height is the shortest distance between the parallel lines. If only a sloping side is given, more information is needed to find the perpendicular height; do not silently treat that side as h.
Undo the operations in a clear order
For area 72 cm² and parallel sides 8 and 16 cm, h = 2 × 72/(8 + 16) = 6 cm. If one parallel side is unknown, first find their sum using a + b = 2A/h, then subtract the known side. Keep brackets around the sum when dividing.
Perimeter still means adding all four boundary lengths. The area formula does not supply an unknown sloping side by itself. State which result can be determined from the available data. A calculated negative height or parallel length is incompatible with an ordinary physical trapezium and should prompt a check of the algebra or given conditions.
Pause and explain
Parallel sides are 6 cm and 10 cm; perpendicular height is 5 cm. Find the area.
Worked example
A trapezium has area 54 cm², height 6 cm and one parallel side 7 cm. Find the other parallel side.
Show the worked solution
- The parallel-side sum is 2 × 54/6 = 18 cm.
- Subtract the known side: 18 − 7 = 11 cm.
- Check (7 + 11) × 6/2 = 54 cm².
Answer 11 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the area of a trapezium with parallel sides 4 cm and 9 cm and height 8 cm.
Give me a hint
Add the parallel sides, then multiply by half the height.
Compare my reasoning
- The parallel-side sum is 13 cm.
- Area is 13 × 8/2 = 52 cm².
- The average width 6.5 cm provides a second check.
52 cm².
Look for these in your work
- Selected the parallel pair.
- Used perpendicular height.
A trapezium has area 90 cm² and parallel sides 12 cm and 18 cm. Find its height.
Give me a hint
Use h = 2A/(a + b).
Compare my reasoning
- The parallel sides total 30 cm.
- h = 180/30 = 6 cm.
- Check 30 × 6/2 = 90 cm².
6 cm.
Look for these in your work
- Rearranged with the factor two.
- Verified the result by substitution.
A plot has parallel boundaries 14 m and x m, separation 9 m and area 153 m². Find x. Can its perimeter be determined from these data alone?
Give me a hint
Find the width sum; distinguish area data from sloping boundary lengths.
Compare my reasoning
- 14 + x = 2 × 153/9 = 34 m.
- x = 20 m.
- The two other boundary lengths are unspecified, so the perimeter is not uniquely determined.
x = 20 m; perimeter cannot be uniquely determined.
Look for these in your work
- Recovered the unknown parallel length.
- Did not invent sloping boundary measurements.
Common mistakes
- Averaging the non-parallel sides.
- Forgetting division by two.
- Claiming the perimeter from insufficient boundary data.
What can you explain now?
Which height belongs in the trapezium area formula?
Compare with the explanation
The perpendicular distance between the parallel sides.
A sloping boundary does not usually measure the shortest separation of the parallel lines.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, recover a missing parallel side from a trapezium's area and height, then substitute to check.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.