Mensuration deals with measurement of geometric figures — both flat (2-D) and solid (3-D) shapes. In SSC MTS Paper 1, expect 2–4 direct questions testing formulas for area, perimeter, surface area, and volume. 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.
**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.
You read the notes — now try one
A rectangular garden is 24 m long and 15 m wide. A path of uniform width 2 m runs around the outside of the garden. What is the area of the path in square metres?
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A rectangular garden is 24 m long and 15 m wide. A path of uniform width 2 m runs around the outside of the garden. What is the area of the path in square metres?
Q2 · Mensuration · MEDIUM
A cylindrical water tank has a radius of 3.5 m and a height of 6 m. What is the capacity of the tank in cubic metres? (Use pi = 22/7)
Q3 · Mensuration · MEDIUM
The length of a rectangle is increased by 20% and its breadth is decreased by 10%. What is the percentage change in its area?
Q4 · Mensuration · HARD
A solid metallic sphere of radius 6 cm is melted and recast into smaller spheres each of radius 2 cm. How many small spheres can be made?
Q5 · Mensuration · EASY
Find the area of a rectangle whose length is 15 cm and breadth is 8 cm.