Area formulae for triangles, parallelograms and trapezia
Three formulae cover most flat shape questions on GCSE Foundation and crossover papers.
Triangle
Area = ½ × base × height.
The "base" is any side; the "height" is the perpendicular distance from the opposite vertex to that base (NOT the slant side).
Worked example: triangle with base 8 cm and perpendicular height 5 cm.
- Area = ½ × 8 × 5 = 20 cm².
Parallelogram
Area = base × perpendicular height.
Same idea — the height is perpendicular, not the slanted side.
Worked example: parallelogram with base 12 cm and perpendicular height 7 cm.
- Area = 12 × 7 = 84 cm².
Trapezium
Area = ½ (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular distance between them.
Worked example: trapezium with parallel sides 6 cm and 10 cm, perpendicular height 4 cm.
- Area = ½ × (6 + 10) × 4 = ½ × 64 = 32 cm².
Why these formulae work
- A parallelogram can be cut and rearranged into a rectangle — same base, same height.
- A triangle is half a parallelogram.
- A trapezium is the average of two parallels times height.
Combined area problems
Real exam questions often combine shapes. Approach:
- Identify each component shape.
- Apply the correct formula.
- Sum (or subtract, if cutting out).
Worked example: an L-shape made of a 4×3 rectangle on top of a 6×3 rectangle.
- Top: 4 × 3 = 12 cm².
- Bottom: 6 × 3 = 18 cm².
- Total: 30 cm².
Triangle from coordinates
For a triangle with vertices on coordinate axes, find the base and perpendicular height directly from coordinates.
⚠Common mistakes
- Using slant height instead of perpendicular height (especially in trapezia and parallelograms).
- Forgetting the ½ in triangle and trapezium formulae.
- Confusing "base" with the longest side — any side can be the base.
- Mismatching units — same units for both length and height.
- Adding the wrong sides in trapezium formula — only the parallel sides go in (a + b).
➜Try this— Quick check
A trapezium has parallel sides 5 cm and 9 cm, height 6 cm. Area?
- ½ × (5 + 9) × 6 = ½ × 14 × 6 = 42 cm².
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