Perimeter & Area • Topic 2 of 3

2: Area of Trapezium and Perimeter Calculations

What is a Trapezium (Trapezoid)?

A trapezium (also called trapezoid) is a quadrilateral with one pair of parallel sides. The parallel sides are called bases (\(a\) and \(b\)), and the perpendicular distance between them is the height (\(h\)).

Area of a Trapezium Formula:

\(A = \frac{1}{2} \times (a + b) \times h\)

Where:

  • \(a\) and \(b\) = lengths of the parallel sides (bases)
  • \(h\) = perpendicular height (distance between the bases)

Why this formula works: A trapezium can be divided into two triangles or transformed into a parallelogram to derive the formula.

Perimeter of a Trapezium:

\(P = a + b + c + d\) (sum of all four sides)

Perimeter of Other Shapes (Review):

ShapePerimeter Formula
Rectangle\(P = 2(l + w)\)
Square\(P = 4s\)
Triangle\(P = a + b + c\)
Parallelogram\(P = 2(a + b)\)
Trapezium\(P = a + b + c + d\)

Special Case - Isosceles Trapezium: Non-parallel sides are equal (\(c = d\)), so \(P = a + b + 2c\)

Real-life Examples of Trapezium:

  • Roof of a house (cross-section)
  • Bridge supports
  • Handbags and purses
  • Table tops
  • Swimming pool cross-section
Area Formulas — Visual ReferenceRectanglel × wTriangle½ b × hParallelogramb × hTrapezium½(a+b)×hCircleπ r²hPerimeter: add all sides | Area: space inside the shapeCircumference of circle = 2πr | Area = πr² | Use π ≈ 3.14 or 22/7
Solved Examples
1
Worked Example
Example 1: Find the area of a trapezium with parallel sides 10 cm and 14 cm, and height 6 cm.
Solution- Step 1: Identify \(a = 10\) cm, \(b = 14\) cm, \(h = 6\) cm - Step 2: Use formula \(A = \frac{1}{2} \times (a + b) \times h\) - Step 3: \(A = \frac{1}{2} \times (10 + 14) \times 6\) - Step 4: \(A = \frac{1}{2} \times 24 \times 6\) - Step 5: \(A = 12 \times 6 = 72\) cm²
2
Worked Example
Example 2: The parallel sides of a trapezium are in the ratio 3:5. The height is 8 cm and area is 96 cm². Find the lengths of the parallel sides.
Solution- Step 1: Let the parallel sides be \(3x\) and \(5x\) - Step 2: Area = \(\frac{1}{2} \times (3x + 5x) \times 8 = 96\) - Step 3: \(\frac{1}{2} \times 8x \times 8 = 96\) - Step 4: \(4x \times 8 = 96\) → \(32x = 96\) - Step 5: \(x = 3\) - Step 6: Sides are \(3x = 9\) cm and \(5x = 15\) cm
3
Worked Example
Example 3: An isosceles trapezium has parallel sides of length 20 cm and 12 cm, and non-parallel sides of length 5 cm each. Find its perimeter and area. (Hint: Find height first using Pythagoras)
Solution- Step 1: Perimeter = \(20 + 12 + 5 + 5 = 42\) cm - Step 2: For area, need height. Difference of bases = \(20 - 12 = 8\) cm - Step 3: This difference is split equally on both sides: \(8 \div 2 = 4\) cm overhang on each side - Step 4: Height forms right triangle with hypotenuse 5 cm and base 4 cm - Step 5: Using Pythagoras: \(h^2 + 4^2 = 5^2\) - Step 6: \(h^2 + 16 = 25\) → \(h^2 = 9\) → \(h = 3\) cm - Step 7: Area = \(\frac{1}{2} \times (20 + 12) \times 3 = \frac{1}{2} \times 32 \times 3 = 16 \times 3 = 48\) cm²

Key Points

  • Trapezium: Quadrilateral with one pair of parallel sides (bases)
  • Area formula: \(A = \frac{1}{2} \times (a + b) \times h\) (average of bases × height)
  • Height must be perpendicular to the bases
  • Perimeter: Sum of all four sides
  • Isosceles trapezium: Non-parallel sides are equal; base angles are equal
  • To find height, sometimes need Pythagoras theorem when given slant height
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The area of a trapezium is:
Explanation: $\tfrac12(a+b)h$.
Q2.The perimeter of a figure is the total length of its:
Explanation: Boundary.
Q3.In a trapezium, the height is the ___ distance between the parallel sides.
Explanation: Perpendicular.
Q4.The perimeter of a square of side $s$ is:
Explanation: $4s$.
Practice more MCQs → 📝 Assignment · 35 marks