Mensuration

Mensuration

Overview

Mensuration is the branch of mathematics dealing with measurement of geometric figures—calculating perimeter, area, and volume. These questions are calculation-heavy but formula-driven, meaning once you memorize the formulas, solving becomes mechanical.

The exam tests your ability to quickly recall the correct formula and perform clean arithmetic under time pressure. Questions often involve finding area when dimensions change, comparing costs of fencing or painting, or calculating the volume of containers. Since IBPS Clerk is a speed-based exam, knowing formulas cold—without hesitation—is your biggest advantage here.

Key Concepts

• Perimeter is the total boundary length of a 2D shape; Area is the space enclosed within that boundary.
• Volume measures the 3D space occupied by a solid; Surface Area measures the total area covering the outer surface.
• For rectangles, length and breadth are different; for squares, all four sides are equal—this distinction affects which formula to use.
• A circle's measurements depend entirely on radius (r); diameter = 2r is often given instead, so always convert first.
• Cylinders combine circular bases with height; their curved surface area excludes the top and bottom circles.
• When dimensions are scaled by a factor k: area scales by k², volume scales by k³.
• Units matter: if dimensions are in cm, area is in cm², volume is in cm³. Convert before calculating if units differ.
• π = 22/7 or 3.14—use whichever simplifies calculation (22/7 when radius is a multiple of 7).

Formulas / Key Facts

2D Shapes

3D Solids

Quick Conversions

• 1 m = 100 cm; 1 m² = 10,000 cm²; 1 m³ = 10,00,000 cm³
• 1 litre = 1000 cm³ = 0.001 m³

Worked Examples

Example 1: Rectangle Area and Cost

Problem: A rectangular garden is 25 m long and 18 m wide. Find the cost of fencing it at ₹12 per metre.

Solution:

• Perimeter = 2(l + b) = 2(25 + 18) = 2 × 43 = 86 m
• Cost = 86 × 12 = ₹1,032

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Example 2: Circle Area

Problem: The radius of a circular park is 14 m. Find its area. (Use π = 22/7)

Solution:

• Area = πr² = (22/7) × 14 × 14
• = (22/7) × 196 = 22 × 28 = 616 m²

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Example 3: Cylinder Volume

Problem: A cylindrical tank has radius 7 cm and height 10 cm. Find the volume of water it can hold.

Solution:

• Volume = πr²h = (22/7) × 7 × 7 × 10
• = (22/7) × 490 = 22 × 70 = 1,540 cm³
• In litres = 1540/1000 = 1.54 litres

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Example 4: Comparing Areas

Problem: If the side of a square is increased by 50%, by what percentage does the area increase?

Solution:

• Let original side = a, new side = 1.5a
• Original area = a², new area = (1.5a)² = 2.25a²
• Increase = 2.25a² − a² = 1.25a²
• Percentage increase = (1.25a²/a²) × 100 = 125%

Common Mistakes

• Confusing perimeter with area → Perimeter is a length (one dimension), area is square units (two dimensions). Read what's asked—fencing needs perimeter, painting/carpet needs area.
• Forgetting to halve diameter → When given diameter, students plug it directly into πr² formulas. Always divide diameter by 2 to get radius first.
• Using wrong surface area formula → CSA means only the curved part (no top/bottom); TSA includes all surfaces. Read whether the question asks for painting the curved surface or the entire tank.
• Unit mismatch errors → If length is in metres and breadth in centimetres, convert both to the same unit before multiplying. Area of 5 m × 200 cm ≠ 1000; convert to 5 m × 2 m = 10 m².
• Scaling errors → When side doubles, area quadruples (not doubles). Remember: area ∝ (side)², volume ∝ (side)³.

Quick Reference

• Square: Perimeter = 4a, Area = a²
• Rectangle: Perimeter = 2(l+b), Area = l×b
• Circle: Circumference = 2πr, Area = πr²
• Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r+h)
• Doubling side → Area becomes 4 times, Volume becomes 8 times
• 1 litre = 1000 cm³; always check unit consistency before solving