Two quadrilaterals that appear regularly at KS3 are the parallelogram and the trapezium. Both have an area formula that depends on the perpendicular height — not the slant side — and learning to identify that difference is the key to answering these questions correctly every time.
What is the area of a parallelogram?
A parallelogram is a four-sided shape with two pairs of parallel sides. Its opposite sides are equal in length, and its opposite angles are equal.
The formula for its area is:
A = base × perpendicular height
or in symbols: A = b × h
The perpendicular height is the straight-line distance between the two parallel bases, measured at a right angle to them. It is not the length of the slant side.
Worked Example 1 — A parallelogram has base 8 cm and perpendicular height 5 cm. Find its area.
- Write the formula: A = b × h
- Substitute: A = 8 × 5
- Calculate: A = 40 cm²
Worked Example 2 — A parallelogram has base 11 cm, a slant side of 7 cm, and a perpendicular height of 6 cm. Find its area.
- Write the formula: A = b × h
- Identify the perpendicular height: h = 6 cm (not the slant side, 7 cm)
- Substitute: A = 11 × 6
- Calculate: A = 66 cm²
Many students mistakenly use A = 11 × 7 = 77 cm² here. The slant side is irrelevant to the area calculation.
Why does base × height work for a parallelogram?
Here is the visual argument that makes the formula memorable. Imagine cutting a right-angled triangle off the left end of the parallelogram and sliding it to the right end. The shape becomes a rectangle with the same base and the same perpendicular height. The area of a rectangle is length × width, which is exactly base × perpendicular height. So the parallelogram formula is just the rectangle formula in disguise.
This also explains why the slant side is irrelevant: it plays no role once you rearrange the pieces into a rectangle.
What is the area of a trapezium?
A trapezium has exactly one pair of parallel sides (called the parallel sides or bases). The other two sides can be any length or angle.
The formula is:
A = ½(a + b) × h
where:
- a and b are the lengths of the two parallel sides
- h is the perpendicular height between them
The ½ appears because two identical trapezia can be placed together to form a parallelogram with base (a + b) and height h, so each trapezium is half that area.
Worked Example 3 — A trapezium has parallel sides of 7 cm and 11 cm, and a perpendicular height of 4 cm. Find its area.
- Write the formula: A = ½(a + b) × h
- Substitute: A = ½ × (7 + 11) × 4
- Add the parallel sides: A = ½ × 18 × 4
- Multiply: A = ½ × 72
- Calculate: A = 36 cm²
Worked Example 4 — A trapezium has parallel sides of 3 m and 9 m, and a perpendicular height of 5 m. Find its area.
- A = ½ × (3 + 9) × 5
- A = ½ × 12 × 5
- A = ½ × 60
- A = 30 m²
How do I remember the trapezium formula?
The phrase "half the sum of the parallel sides, times the height" gives you the formula word for word:
- Half → ½
- sum of the parallel sides → (a + b)
- times the height → × h
Alternatively, think of the trapezium as the average of a rectangle with width a and a rectangle with width b, both with height h:
Average width = (a + b) ÷ 2, so Area = average width × height = ½(a + b) × h.
Both routes lead to exactly the same formula.
Can I work backwards to find the height?
Yes. Rearrange A = ½(a + b) × h to make h the subject:
- Multiply both sides by 2: 2A = (a + b) × h
- Divide both sides by (a + b): h = 2A ÷ (a + b)
Worked Example 5 — A trapezium has area 48 cm² and parallel sides of 6 cm and 10 cm. Find the perpendicular height.
- h = 2 × 48 ÷ (6 + 10)
- h = 96 ÷ 16
- h = 6 cm
Check: A = ½ × (6 + 10) × 6 = ½ × 16 × 6 = ½ × 96 = 48 cm² ✓
What are the most common mistakes?
Mistake 1: Using the slant side instead of the perpendicular height.
Both the parallelogram and the trapezium formulae require the perpendicular height — the distance measured at 90° to the parallel sides. If a diagram shows a slant side of 7 cm but the perpendicular height is 6 cm, use 6 cm.
Mistake 2: Adding all four sides of the trapezium instead of just the two parallel ones.
Only the two parallel sides, a and b, go into the trapezium formula. The non-parallel sides (the legs) do not feature in the area calculation; they would be relevant only if you were finding the perimeter.
Mistake 3: Forgetting the ½ in the trapezium formula.
Writing A = (a + b) × h instead of A = ½(a + b) × h doubles the answer. The factor of a half is essential.
| Shape | Formula | Key variables |
|---|---|---|
| Parallelogram | A = b × h | b = base; h = perpendicular height |
| Trapezium | A = ½(a + b) × h | a, b = parallel sides; h = perpendicular height |
Frequently asked questions
Is a parallelogram the same as a rectangle?
A rectangle is a special case of a parallelogram where all four angles are 90°. Every rectangle is a parallelogram, but not every parallelogram is a rectangle. The area formula A = b × h applies to both, because a rectangle's perpendicular height equals its width side.
Do both pairs of sides have to be parallel in a trapezium?
In the UK definition, a trapezium has exactly one pair of parallel sides. A shape with two pairs of parallel sides is a parallelogram (or one of its special cases: rectangle, rhombus, square). This is worth knowing because the trapezium formula only applies to shapes with one pair of parallel sides.
Why do I need to identify the parallel sides carefully in a trapezium?
The formula adds the two parallel sides. If you accidentally use one parallel side and one non-parallel side, your sum (a + b) will be wrong and so will the area. Always check: which two sides are parallel? They are often the top and bottom of the shape in a diagram, but not always — exam diagrams sometimes tilt the shape to test whether you can identify them correctly.
How does the trapezium formula relate to the triangle area formula?
The triangle formula is A = ½ × base × height. A triangle is a trapezium where one of the parallel sides has length zero, so setting a = 0 in the trapezium formula gives A = ½(0 + b) × h = ½bh — exactly the triangle formula. This is a neat way to see how the two formulae are connected.
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