Area of a trapezium, explained with examples
The area of a trapezium is half the sum of the parallel sides, multiplied by the height. It is one of the formulas you must know for GCSE Maths, and it turns up in compound shapes, prisms and even calculus.
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What a trapezium is
A trapezium is a four-sided shape with exactly one pair of parallel sides. The parallel sides, often labelled a and b, can be different lengths. The height h is the perpendicular distance between the parallel sides, not the length of a sloping side. (In the US, this shape is called a trapezoid.)
The formula
Area = ½(a + b)h
In words: add the parallel sides, halve the total, and multiply by the height. An equivalent way to think of it is the average of the parallel sides times the height. Whether it is printed on a formula sheet in your exam depends on the board and the year, so the safest approach is to learn it until it is automatic.
Why the formula works
There are two neat proofs. First, take two identical copies of the trapezium, turn one upside down, and place them side by side. Together they make a parallelogram with base a + b and height h, so its area is (a + b)h. One trapezium is half of this: ½(a + b)h.
Second, draw a diagonal to split the trapezium into two triangles. One has base a and height h, with area ½ah; the other has base b and height h, with area ½bh. Adding them gives ½ah + ½bh = ½(a + b)h.
Notice what happens in special cases. If a = b, the shape is a parallelogram and the formula gives ah. If a = 0, the shape becomes a triangle and the formula gives ½bh. The trapezium formula contains both.
Worked examples
Example 1: the basic formula
Find the area of a trapezium with parallel sides 6 cm and 10 cm and height 5 cm.
- ½(6 + 10) × 5 = ½ × 16 × 5.
- Area = 40 cm².
Example 2: decimals
Find the area of a trapezium with parallel sides 4.5 m and 7.3 m and height 3.2 m.
- ½(4.5 + 7.3) × 3.2 = 5.9 × 3.2.
- Area = 18.88 m².
Example 3: finding the height
A trapezium has area 54 cm² and parallel sides 7 cm and 11 cm. Find its height.
- 54 = ½(7 + 11)h = 9h.
- h = 6 cm.
Example 4: finding a parallel side
A trapezium has area 60 cm², height 8 cm and one parallel side 9 cm. Find the other parallel side.
- 60 = ½(9 + b) × 8 = 4(9 + b), so 9 + b = 15.
- b = 6 cm.
Example 5: an isosceles trapezium (Higher)
An isosceles trapezium has parallel sides 6 cm and 14 cm and sloping sides 5 cm. Find its area.
- The difference in parallel sides is 8 cm, shared equally: 4 cm on each side. Each sloping side forms a right-angled triangle with base 4 cm and hypotenuse 5 cm.
- Height = √(5² − 4²) = 3 cm.
- Area = ½(6 + 14) × 3 = 30 cm².
Example 6: units
A trapezium has parallel sides 80 cm and 1.2 m, and height 50 cm. Find its area in m².
- Convert first: 0.8 m, 1.2 m and 0.5 m.
- ½(0.8 + 1.2) × 0.5 = 0.5 m².
Compound shapes
Trapeziums often appear inside L-shapes, house outlines or garden plans. Split the shape into rectangles, triangles and trapeziums, find each area, and add. Sometimes it is quicker to find a large shape and subtract a missing piece.
Example 7: a house end
The end wall of a shed is a rectangle 3 m wide and 2 m tall, with a trapezium-shaped roof section on top whose parallel sides are 3 m and 1 m, with height 0.8 m. Find the total area.
- Rectangle: 3 × 2 = 6 m². Trapezium: ½(3 + 1) × 0.8 = 1.6 m².
- Total = 7.6 m².
Trapezium prisms
A prism whose cross-section is a trapezium, such as a swimming pool that gets deeper from one end to the other, has volume equal to the area of the trapezium times the length.
Example 8: a swimming pool
A pool is 25 m long and 10 m wide. Its depth increases steadily from 1 m at the shallow end to 2.5 m at the deep end. Find its volume and capacity.
- The side view is a trapezium with parallel sides 1 m and 2.5 m and “height” 25 m (the length).
- Area = ½(1 + 2.5) × 25 = 43.75 m². Volume = 43.75 × 10 = 437.5 m³.
- Capacity = 437.5 × 1000 = 437,500 litres.
The trapezium rule (A level link)
At A level, the same idea estimates the area under a curve. The region is split into vertical strips, and each strip is treated as a trapezium. Adding them up gives the trapezium rule, a simple and powerful numerical method. The GCSE formula you are learning now is the building block.
Common mistakes
- Using the length of a sloping side instead of the perpendicular height.
- Forgetting to halve, or halving twice.
- Adding the height to the parallel sides instead of multiplying.
- Mixing units, such as cm and m in the same calculation.
Practice questions
Try each one before opening the answer.
Q1. Find the area of a trapezium with parallel sides 3 cm and 9 cm and height 4 cm.
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Q2. A trapezium has area 35 cm², height 5 cm and one parallel side 5 cm. Find the other.
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Q3. Show that a trapezium with parallel sides both 8 cm and height 3 cm has area 24 cm².
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Q4. A trapezium prism has cross-section area 12 cm² and length 15 cm. Find its volume.
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Frequently asked questions
Is the trapezium formula given in the GCSE exam?
It depends on your exam board and the exam series, as formula sheet arrangements have changed in recent years. Check what your board provides, but learn it anyway: knowing it saves time and avoids mistakes.
What is the difference between a trapezium and a trapezoid?
In the UK, a trapezium has one pair of parallel sides. In the US, that shape is called a trapezoid. The formula is the same.
Key takeaways
- Area = ½(a + b)h, where a and b are the parallel sides.
- h is the perpendicular height, not a sloping side.
- Two trapeziums make a parallelogram, which proves the formula.
- Rearrange to find a missing side or height.
- Prism volume = trapezium area × length.