Area of a Trapezium

\( A = \tfrac{1}{2}(a+b)h \)
Geometry GCSE
Question 2 of 20
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Find the area of a trapezium with parallel sides 8 cm and 14 cm, height 7 cm.

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( Use A = 1/2 (a+b)h \)

Explanation

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Statement

The area of a trapezium (a quadrilateral with one pair of parallel sides) can be calculated using the lengths of the parallel sides and the perpendicular height. The formula is:

\[ A = \tfrac{1}{2}(a+b)h \]

Here, \(a\) and \(b\) are the lengths of the parallel sides, and \(h\) is the perpendicular height between them.

Why it’s true (short reason)

  • The trapezium can be split into a rectangle and two right-angled triangles, whose areas add up to this formula.
  • Alternatively, two congruent trapezia can be joined to form a parallelogram with base \((a+b)\) and height \(h\). The area of the trapezium is then half of that parallelogram.

Recipe (how to use it)

  1. Identify the two parallel sides, \(a\) and \(b\).
  2. Measure or calculate the perpendicular height \(h\).
  3. Add the parallel sides: \(a+b\).
  4. Multiply by the height: \((a+b)h\).
  5. Halve the result: \(A = \tfrac{1}{2}(a+b)h\).

Spotting it

You should apply this formula when:

  • The quadrilateral has one pair of parallel sides.
  • The problem gives both parallel side lengths and the perpendicular height.
  • The question asks directly for trapezium area or provides enough information to deduce it.

Common pairings

  • Right-angled triangle area formulas when breaking trapezia into parts.
  • Pythagoras’ theorem to calculate the height if not given directly.
  • Coordinate geometry, where trapezia are often defined on axes.

Mini examples

  1. Given: A trapezium with parallel sides 8 cm and 12 cm, height 6 cm. Find: Area. Answer: \(A = \tfrac{1}{2}(8+12)\times 6 = 60\ \text{cm}^2\).
  2. Given: A trapezium with parallel sides 15 m and 20 m, height 10 m. Find: Area. Answer: \(A = \tfrac{1}{2}(15+20)\times 10 = 175\ \text{m}^2\).

Pitfalls

  • Using slanted sides instead of the perpendicular height.
  • Forgetting to halve the product.
  • Confusing trapezium with parallelogram — a parallelogram’s area is \(bh\), not \(\tfrac{1}{2}(a+b)h\).
  • Mixing units (cm with mm, etc.) without conversion.

Exam strategy

  • Underline “parallel sides” in the question.
  • Draw the perpendicular height if it is not marked in the diagram.
  • Check carefully whether the height is given or must be found using Pythagoras or trigonometry.
  • Always write the formula before substitution to avoid missing the \(\tfrac{1}{2}\).

Summary

The trapezium area formula \(A = \tfrac{1}{2}(a+b)h\) comes from combining the parallel sides and multiplying by the perpendicular height, then halving. It works for all trapezia, including isosceles and right-angled cases. Always identify the parallel sides, check the height is perpendicular, and calculate carefully with consistent units.

Worked examples

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  1. Find the area of a trapezium with parallel sides 10 cm and 6 cm, and height 5 cm.
    1. \( A = 1/2 (10+6) × 5 \)
    2. \( A = 1/2 × 16 × 5 \)
    3. \( A = 40 \)
    Answer: \( 40\ \text{cm}^2 \)
  2. A trapezium has parallel sides 15 m and 25 m, height 8 m. Find its area.
    1. \( A = 1/2 (15+25) × 8 \)
    2. \( A = 1/2 × 40 × 8 \)
    3. \( A = 160 \)
    Answer: \( 160\ \text{m}^2 \)
  3. Find the area of a trapezium with parallel sides 9 cm and 11 cm, and height 7 cm.
    1. \( A = 1/2 (9+11) × 7 \)
    2. \( A = 1/2 × 20 × 7 \)
    3. \( A = 70 \)
    Answer: \( 70\ \text{cm}^2 \)
  4. A trapezium has parallel sides 12 cm and 18 cm, height 10 cm.
    1. \( A = 1/2 (12+18) × 10 \)
    2. \( A = 1/2 × 30 × 10 \)
    3. \( A = 150 \)
    Answer: \( 150\ \text{cm}^2 \)
  5. A trapezium has parallel sides 20 m and 30 m, height 12 m. Find the area.
    1. \( A = 1/2 (20+30) × 12 \)
    2. \( A = 1/2 × 50 × 12 \)
    3. \( A = 300 \)
    Answer: \( 300\ \text{m}^2 \)
  6. Find the area of a trapezium with parallel sides 14 cm and 22 cm, and height 9 cm.
    1. \( A = 1/2 (14+22) × 9 \)
    2. \( A = 1/2 × 36 × 9 \)
    3. \( A = 162 \)
    Answer: \( 162\ \text{cm}^2 \)
  7. A trapezium has parallel sides 7 m and 13 m, height 10 m.
    1. \( A = 1/2 (7+13) × 10 \)
    2. \( A = 1/2 × 20 × 10 \)
    3. \( A = 100 \)
    Answer: \( 100\ \text{m}^2 \)
  8. Find the area of a trapezium with parallel sides 25 cm and 35 cm, and height 15 cm.
    1. \( A = 1/2 (25+35) × 15 \)
    2. \( A = 1/2 × 60 × 15 \)
    3. \( A = 450 \)
    Answer: \( 450\ \text{cm}^2 \)
  9. A trapezium has parallel sides 40 cm and 50 cm, height 20 cm.
    1. \( A = 1/2 (40+50) × 20 \)
    2. \( A = 1/2 × 90 × 20 \)
    3. \( A = 900 \)
    Answer: \( 900\ \text{cm}^2 \)
  10. Find the area of a trapezium with parallel sides 18 cm and 24 cm, and height 11 cm.
    1. \( A = 1/2 (18+24) × 11 \)
    2. \( A = 1/2 × 42 × 11 \)
    3. \( A = 231 \)
    Answer: \( 231\ \text{cm}^2 \)