Mensuration — SSC MTS Study Notes
Overview
Mensuration deals with measurement of geometric figures — both flat (2-D) and solid (3-D) shapes. Most questions are straightforward formula substitution, though some involve unit conversion or composite figures.
This topic is a score-booster because formulas are fixed and calculation steps are mechanical. Master the 10–12 core formulas, practice unit conversions (cm to m, m² to cm²), and recognize when a problem involves combining multiple shapes. Accuracy in applying the correct formula and careful arithmetic are the keys to full marks.
Common question types: find the area of a triangle given base and height; calculate the cost of painting four walls of a room; determine how much water a cylindrical tank holds; find the diagonal of a rectangle or cuboid. Questions may be direct or wrapped in a one-step word problem.
Key Concepts
- 2-D figures have only length and width — we measure their perimeter (boundary length) and area (surface covered). Common shapes: square, rectangle, triangle, circle, trapezium.
- 3-D solids have length, width, and height — we measure surface area (total outer surface) and volume (space occupied). Common solids: cube, cuboid, cylinder, cone, sphere, hemisphere.
- Perimeter is always in linear units (cm, m) while area is in square units (cm², m²). Volume is in cubic units (cm³, m³, litres where 1 litre = 1000 cm³).
- Composite figures: Some problems combine shapes — e.g. a rectangle with a semicircle on top. Break the figure into known shapes, calculate separately, then add or subtract as needed.
- Unit conversion: 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Always convert to the same unit before calculation.
- Diagonal formulas: For rectangle/cuboid, use Pythagoras theorem. Rectangle diagonal = √(l² + b²). Cuboid diagonal = √(l² + b² + h²).
- In cost problems, Total Cost = (Area or Volume) × Rate per unit. E.g., cost of flooring = floor area × rate per m².
Formulas / Key Facts
2-D Figures:
- Rectangle: Perimeter = 2(l + b); Area = l × b; Diagonal = √(l² + b²)
- Square: Perimeter = 4a; Area = a²; Diagonal = a√2 (where a = side)
- Triangle: Perimeter = a + b + c (sum of sides); Area = ½ × base × height. For equilateral triangle of side a: Area = (√3/4)a²
- Circle: Circumference = 2πr or πd; Area = πr² (where r = radius, d = diameter, π ≈ 22/7 or 3.14)
- Semicircle: Perimeter = πr + 2r (curved part + diameter); Area = (πr²)/2
- Trapezium: Area = ½ × (sum of parallel sides) × height = ½(a + b)h
- Parallelogram: Area = base × height
3-D Solids:
- Cube: Total Surface Area = 6a²; Lateral Surface Area = 4a²; Volume = a³; Diagonal = a√3
- Cuboid: Total Surface Area = 2(lb + bh + hl); Lateral Surface Area = 2h(l + b); Volume = l × b × h; Diagonal = √(l² + b² + h²)
- Cylinder: Curved Surface Area = 2πrh; Total Surface Area = 2πr(r + h); Volume = πr²h
- Cone: Curved Surface Area = πrl (l = slant height = √(r² + h²)); Total Surface Area = πr(r + l); Volume = (1/3)πr²h
- Sphere: Surface Area = 4πr²; Volume = (4/3)πr³
- Hemisphere: Curved Surface Area = 2πr²; Total Surface Area = 3πr²; Volume = (2/3)πr³
Worked Examples
Example 1: A rectangular field is 50 m long and 30 m wide. Find the cost of fencing it at ₹12 per metre.
Solution: Perimeter = 2(l + b) = 2(50 + 30) = 2 × 80 = 160 m. Cost = 160 × 12 = ₹1920.
Example 2: Find the area of a circle whose circumference is 44 cm. (Use π = 22/7)
Solution: Circumference = 2πr = 44. So 2 × (22/7) × r = 44 → (44/7) × r = 44 → r = 7 cm. Area = πr² = (22/7) × 7 × 7 = 154 cm².
Example 3: A cylindrical water tank has radius 1.4 m and height 2 m. How many litres of water can it hold? (Use π = 22/7)
Solution: Volume = πr²h = (22/7) × 1.4 × 1.4 × 2 = (22/7) × 1.96 × 2 = 22 × 0.28 × 2 = 12.32 m³. Converting to litres: 1 m³ = 1000 litres, so 12.32 m³ = 12,320 litres.
Example 4: A room is 5 m long, 4 m wide and 3 m high. Find the cost of painting its four walls at ₹8 per m².
Solution: Lateral Surface Area of cuboid = 2h(l + b) = 2 × 3 × (5 + 4) = 6 × 9 = 54 m². Cost = 54 × 8 = ₹432.
Common Mistakes
- Using perimeter formula for area: Students confuse 2(l + b) (perimeter) with l × b (area) for rectangles. Fix: Remember perimeter measures boundary, area measures surface.
- Forgetting to square or cube units: Writing area in cm instead of cm², or volume in m instead of m³. Fix: Check dimensionality — area is always squared units, volume is cubed.
- Wrong unit conversion: Thinking 1 m² = 100 cm² instead of 10,000 cm². Fix: Remember 1 m = 100 cm, so 1 m² = 100 cm × 100 cm = 10,000 cm². Always square or cube the conversion factor.
- Misapplying π approximation: Using π = 22/7 when the problem gives decimals or vice versa. Fix: Use π = 22/7 when radius or diameter is a multiple of 7; use 3.14 otherwise, or follow the question's instruction.
- Confusing total and lateral surface area: Using total surface area when the question asks for curved/lateral (e.g., cost of painting walls excludes floor and ceiling). Fix: Read carefully — "four walls" means lateral surface area only.
- Omitting slant height in cone problems: Using height h instead of slant height l in curved surface area = πrl. Fix: Calculate l = √(r² + h²) first if not given.
Quick Reference
- Rectangle: Area = l × b, Perimeter = 2(l + b), Diagonal = √(l² + b²)
- Circle: Area = πr², Circumference = 2πr
- Cube: Volume = a³, Surface Area = 6a²
- Cylinder: Volume = πr²h, Curved Surface = 2πrh
- Cone: Volume = (1/3)πr²h, slant height l = √(r² + h²)
- 1 m² = 10,000 cm²; 1 m³ = 1,000,000 cm³; 1 litre = 1000 cm³
- For composite shapes, break into parts and add/subtract areas or volumes.