Area of Trapezium and Polygons
1. Area of a Trapezium
A trapezium is a quadrilateral with ONE pair of parallel sides.
Formula: Area = ½ × (sum of parallel sides) × height Area = ½ × (a + b) × h
Where a and b are the lengths of the two PARALLEL sides and h is the PERPENDICULAR distance between them.
Worked Example: Find the area of a trapezium whose parallel sides are 12 cm and 8 cm and the distance between them is 5 cm.
Area = ½ × (12 + 8) × 5 = ½ × 20 × 5 = 50 cm²
Worked Example: The area of a trapezium is 90 cm² and the distance between parallel sides is 6 cm. If one parallel side is 12 cm, find the other.
Area = ½ × (a + b) × h 90 = ½ × (12 + b) × 6 90 = 3(12 + b) 30 = 12 + b b = 18 cm
2. Area of a Rhombus
Formula 1: Using Diagonals
Area = ½ × d₁ × d₂ (where d₁ and d₂ are diagonals)
Formula 2: Using Base and Height
Area = Base × Height (a rhombus is a parallelogram)
Worked Example: Find the area of a rhombus whose diagonals are 12 cm and 16 cm.
Area = ½ × 12 × 16 = 96 cm²
Side of rhombus = √((d₁/2)² + (d₂/2)²) = √(6² + 8²) = √100 = 10 cm
Worked Example: The area of a rhombus is 120 cm² and one diagonal is 15 cm. Find the other diagonal.
120 = ½ × 15 × d₂ 120 = 7.5 × d₂ d₂ = 16 cm
3. Area of a General Quadrilateral
Area = ½ × (length of diagonal) × (sum of perpendiculars from opposite vertices to the diagonal)
Area = ½ × d × (h₁ + h₂)
Worked Example: Find the area of quadrilateral ABCD where diagonal AC = 12 cm, and perpendicular distances from B and D to AC are 4 cm and 6 cm.
Area = ½ × 12 × (4 + 6) = 6 × 10 = 60 cm²
4. Area of Regular Polygons
Regular Pentagon
A regular pentagon can be divided into 5 isosceles triangles. Area = (5/4) × s² × cot(180°/5) or Area = (5/2) × a × s (where a = apothem)
Regular Hexagon
A regular hexagon can be divided into 6 equilateral triangles. Area = (3√3/2) × s²
Worked Example: Find the area of a regular hexagon with side 4 cm.
Area = (3√3/2) × 4² = (3√3/2) × 16 = 24√3 cm² ≈ 41.57 cm²
5. Area of Irregular Polygons
Method: Divide the irregular polygon into triangles, trapeziums, and rectangles. Find the area of each. ADD them.
Worked Example: Find the area of an irregular pentagon ABCDE where: AC = 10 cm, BF = 3 cm (perpendicular from B to AC) AG = 5 cm (perpendicular from A to CD extension) DH = 4 cm (perpendicular from D to AC) EI = 2 cm (perpendicular from E to AC extension)
Area = Area of triangle ABC + Area of triangle ACD + Area of triangle ADE Area = ½ × 10 × 3 + ½ × 10 × 4 + ½ × 10 × 2 Area = 15 + 20 + 10 = 45 cm²
6. Real-Life Applications
Worked Example: A trapezium-shaped park has parallel sides 40 m and 30 m. The distance between them is 20 m. Find the cost of levelling the park at Rs 15 per m².
Area = ½ × (40 + 30) × 20 = ½ × 70 × 20 = 700 m² Cost = 700 × 15 = Rs 10,500
Worked Example: A rhombus-shaped field has diagonals 24 m and 18 m. Find the cost of fencing it at Rs 20 per metre.
Area = ½ × 24 × 18 = 216 m² Side = √(12² + 9²) = √(144 + 81) = √225 = 15 m Perimeter = 4 × 15 = 60 m Cost = 60 × 20 = Rs 1200
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Using the SLANT height instead of perpendicular height in trapezium' | Height h is the PERPENDICULAR distance between parallel sides, NOT the slant sides |
| 'Using side length as height for rhombus area' | Rhombus area = ½ × d₁ × d₂ (diagonals), or Base × Perpendicular Height |
| 'Forgetting to divide by 2 in the trapezium formula' | Trapezium = ½ × (sum of parallel sides) × height. The ½ is ESSENTIAL |
| 'Counting diagonal length without perpendicular heights' | Quadrilateral area requires the diagonal AND both perpendicular heights |
ICSE Exam Focus (5–7 marks)
- 2-mark questions: Find area of trapezium/rhombus given dimensions
- 3-mark questions: Find missing dimension given area and other dimensions
- 4-mark questions: Irregular polygon area by splitting into simpler shapes
- 6-mark questions: Composite problems with cost calculations
Self-Test
Q1. Find the area of a trapezium with parallel sides 15 cm and 9 cm and height 6 cm. A1. Area = ½ × (15 + 9) × 6 = ½ × 24 × 6 = 72 cm².
Q2. The area of a rhombus is 96 cm² and one diagonal is 12 cm. Find the other diagonal. A2. 96 = ½ × 12 × d₂ → 96 = 6 × d₂ → d₂ = 16 cm.
Q3. Find the area of a quadrilateral whose diagonal is 10 cm and perpendiculars from opposite vertices are 5 cm and 3 cm. A3. Area = ½ × 10 × (5 + 3) = 5 × 8 = 40 cm².
Q4. The parallel sides of a trapezium are 20 m and 14 m. Its area is 170 m². Find the distance between the parallel sides. A4. 170 = ½ × (20 + 14) × h → 170 = 17 × h → h = 10 m.
Q5. A rhombus has diagonals 30 cm and 40 cm. Find its side, perimeter, and area. A5. Side = √(15² + 20²) = √625 = 25 cm. Perimeter = 100 cm. Area = ½ × 30 × 40 = 600 cm².
Q6. Find the area of a regular hexagon with side 6 cm. A6. Area = (3√3/2) × 36 = 54√3 cm² ≈ 93.53 cm².
