Mensuration — Class 8 Mathematics
1. Area of Plane Figures — Quick Recap
| Shape | Area Formula | Perimeter Formula |
|---|---|---|
| Square (side = a) | a² | 4a |
| Rectangle (l×b) | l × b | 2(l+b) |
| Triangle | ½ × base × height | Sum of 3 sides |
| Circle (radius r) | πr² | 2πr (circumference) |
| Parallelogram | base × height | 2(adjacent sides sum) |
| Rhombus | ½ × d₁ × d₂ (product of diagonals) | 4 × side |
2. Area of a Trapezium
A trapezium has ONE pair of parallel sides. Area = ½ × (sum of parallel sides) × height. A = ½(a+b)h, where a and b are the parallel sides and h is the perpendicular distance between them.
Derivation: Split the trapezium into two triangles by drawing a diagonal. Area of triangle 1 = ½×a×h. Area of triangle 2 = ½×b×h. Total = ½h(a+b).
Worked Examples
Example 1: A trapezium has parallel sides 12 cm and 8 cm. Height = 5 cm. Area = ½(12+8)×5 = ½×20×5 = 50 cm². Example 2: Area = 72 cm². Parallel sides: 10 cm and x. Height = 6 cm. 72 = ½(10+x)×6 → 72 = 3(10+x) → 24 = 10+x → x = 14 cm.
3. Area of a General Quadrilateral
For any quadrilateral: draw a diagonal. This splits it into TWO triangles. Area = area of triangle 1 + area of triangle 2 = ½×d×h₁ + ½×d×h₂ = ½d(h₁+h₂), where d = diagonal length, h₁ and h₂ = perpendicular distances from the other two vertices to this diagonal.
Worked Example
A quadrilateral has diagonal 16 cm. Perpendiculars from the remaining vertices are 5 cm and 7 cm. Area = ½×16×(5+7) = 8×12 = 96 cm².
4. Area of a Rhombus
A rhombus is a parallelogram with ALL sides equal. Area = ½ × d₁ × d₂ (half the product of its two diagonals). This is derived from the general quadrilateral formula — in a rhombus, diagonals are perpendicular.
Example: A rhombus has diagonals 8 cm and 6 cm. Area = ½×8×6 = 24 cm². Side length: Since diagonals are perpendicular bisectors, side² = (d₁/2)² + (d₂/2)² = 4²+3² = 25. Side = 5 cm.
5. Area of a Polygon
For a regular polygon: divide it into n congruent isosceles triangles from the centre. Area = n × (area of one triangle) = ½ × perimeter × apothem (apothem = perpendicular from centre to any side).
For an irregular polygon: divide into triangles/quadrilaterals through diagonals. Find area of each shape. Sum them.
6. Surface Area and Volume of 3D Solids
Cuboid (length l, breadth b, height h)
| Measure | Formula |
|---|---|
| LSA (Lateral Surface Area — 4 walls) | 2(l+b)h |
| TSA (Total Surface Area) | 2(lb + bh + hl) |
| Volume | l × b × h |
| Diagonal | √(l²+b²+h²) |
Cube (side = a)
| Measure | Formula |
|---|---|
| LSA | 4a² |
| TSA | 6a² |
| Volume | a³ |
| Diagonal | a√3 |
Cylinder (radius r, height h)
| Measure | Formula |
|---|---|
| CSA (Curved Surface Area) | 2πrh |
| TSA | 2πr(r+h) = 2πr²+2πrh |
| Volume | πr²h |
7. Worked Examples — 3D Solids
Example 1 — Cuboid: A room is 6 m × 4 m × 3 m. Area of 4 walls = 2(l+b)h = 2(6+4)×3 = 60 m². Cost of whitewashing at ₹20/m² = 60×20 = ₹1200. TSA (ceiling + walls + floor) = 2(24+12+18) = 108 m². Volume = 6×4×3 = 72 m³.
Example 2 — Cube: TSA of cube = 96 cm². Find side. 6a² = 96 → a² = 16 → a = 4 cm. Volume = a³ = 64 cm³.
Example 3 — Cylinder: Cylindrical tank, r = 1.4 m, h = 2 m. (Use π = 22/7). CSA = 2πrh = 2×(22/7)×1.4×2 = 17.6 m². TSA = 2πr(r+h) = 2×(22/7)×1.4×3.4 = 29.92 m². Volume = πr²h = (22/7)×1.4²×2 = 12.32 m³ = 12,320 L (1 m³ = 1000 L).
8. Volume and Capacity
Volume = amount of space an object occupies. Capacity = amount of substance (usually liquid) a container can hold. Units: 1 cm³ = 1 mL. 1 m³ = 1000 L. 1 L = 1000 cm³. 'Volume is about the OBJECT. Capacity is about what the object can HOLD.'
9. Common Mistakes
- Area of trapezium using slant height: Use PERPENDICULAR height, not the slant (non-parallel) side.
- TSA vs LSA: LSA excludes TOP and BOTTOM faces. TSA includes EVERYTHING. Read the question carefully.
- Using diameter instead of radius: Formulas use r (radius). If diameter is given, HALVE it first.
- Unit mixing: Convert all dimensions to SAME unit before applying formula.
10. AP Exam Focus
| Topic | Marks |
|---|---|
| Area of trapezium | 3-4 |
| Area of rhombus/quadrilateral | 2-3 |
| TSA and Volume of cuboid/cube | 3-4 |
| Cylinder problems | 4-5 |
Key Exam Tips
- Draw a diagram for EVERY mensuration problem. Label given dimensions.
- Area units: cm², m². Volume units: cm³, m³. NEVER mix unit types.
- For cylinder problems: CSA uses 2πrh. One circle = πr². TSA = CSA + 2 circles.
Quick Self-Test
- Area of trapezium with parallel sides 15 cm, 9 cm and height 6 cm? (Answer: ½×24×6 = 72 cm².)
- Rhombus with diagonals 10 cm and 24 cm — area? (Answer: ½×10×24 = 120 cm².)
- Volume of cylinder with r=7cm, h=10cm? (Answer: (22/7)×49×10 = 1540 cm³.)
- TSA of cube of side 5 cm? (Answer: 6×25 = 150 cm².)
- How many litres in 2.5 m³? (Answer: 2500 L.)
