Mensuration

Study Notes: Mensuration

SSC General Duty Constable — Elementary Mathematics

Overview

This topic tests your ability to calculate areas, perimeters, surface areas, and volumes of common two-dimensional and three-dimensional shapes. Most problems are formula-based with straightforward numerical substitution, though some require two-step reasoning (for example, finding the radius first before calculating area).

For SSC GD, you must be thoroughly familiar with triangles, quadrilaterals (especially rectangles, squares, parallelograms, and trapeziums), circles, and common 3-D solids like cubes, cuboids, cylinders, cones, and spheres. Questions are often set in practical contexts—painting walls, filling tanks, fencing fields, or wrapping boxes—which makes unit awareness crucial. A single formula error or unit mismatch can cost you marks, so accuracy with standard formulas is non-negotiable.

The key to scoring well in mensuration is memorizing the correct formulas, practicing unit conversions (metres to centimetres, square metres to square centimetres), and drilling enough problems to recognize which formula applies instantly. Since SSC GD allows calculators in some shifts (verify your specific instructions), speed matters as much as accuracy.

Key Concepts

• Area measures the surface enclosed by a 2-D figure, expressed in square units (cm², m², etc.). Perimeter measures the boundary length, expressed in linear units (cm, m, etc.).
• Surface area of a 3-D object is the total area covering all its outer faces. Volume measures the space inside a 3-D object, expressed in cubic units (cm³, m³, litres).
• For composite figures (e.g., a path around a rectangle, or a figure made of a rectangle and semicircle), break the shape into simpler parts and add or subtract areas as needed.
• Diagonal of a rectangle = √(length² + breadth²). Diagonal of a square with side a = a√2. These often appear as intermediate steps in multi-step problems.
• The ratio of the circumference of a circle to its diameter is π (pi), approximately 22/7 or 3.14. Most SSC GD problems use π = 22/7 unless stated otherwise.
• For cylinders, cones, and spheres, radius (r) is half the diameter. Confusing radius with diameter is a frequent source of error—always double-check what the question provides.
• When a problem mentions "four walls" of a room, you calculate the lateral surface area (exclude floor and ceiling). If it says "whole room," include all six faces.
• Unit conversions: 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Also, 1 litre = 1000 cm³. Always convert to a common unit before calculating.

Formulas / Key Facts

Two-Dimensional Figures

• Rectangle: Area = length × breadth; Perimeter = 2(length + breadth)
• Square: Area = side²; Perimeter = 4 × side; Diagonal = side√2
• Triangle: Area = ½ × base × height; Perimeter = sum of all three sides
• Equilateral triangle (side a): Area = (√3/4) × a²; Height = (√3/2) × a
• Right triangle: Area = ½ × product of two perpendicular sides; Hypotenuse = √(base² + height²) by Pythagoras
• Parallelogram: Area = base × height; Perimeter = 2(sum of adjacent sides)
• Trapezium: Area = ½ × (sum of parallel sides) × height
• Rhombus: Area = ½ × product of diagonals; also Area = base × height
• Circle (radius r): Area = πr²; Circumference = 2πr = πd (where d = diameter)
• Semicircle: Area = (πr²)/2; Perimeter = πr + 2r (curved part plus diameter)
• Sector of circle (angle θ in degrees): Area = (θ/360) × πr²; Arc length = (θ/360) × 2πr

Three-Dimensional Figures

• Cube (side a): Volume = a³; Total Surface Area = 6a²; Lateral Surface Area = 4a²
• Cuboid (length l, breadth b, height h): Volume = l × b × h; Total Surface Area = 2(lb + bh + hl); Lateral Surface Area = 2h(l + b)
• Cylinder (radius r, height h): Volume = πr²h; Curved Surface Area = 2πrh; Total Surface Area = 2πrh + 2πr² = 2πr(h + r)
• Cone (radius r, height h, slant height l): Volume = (1/3)πr²h; Curved Surface Area = πrl; Total Surface Area = πrl + πr² = πr(l + r); Slant height l = √(r² + h²)
• Sphere (radius r): Volume = (4/3)πr³; Surface Area = 4πr²
• Hemisphere (radius r): Volume = (2/3)πr³; Curved Surface Area = 2πr²; Total Surface Area = 3πr²

Worked Examples

Example 1: Rectangle Area and Perimeter A rectangular field is 50 m long and 30 m wide. Find (a) its area, (b) perimeter, and (c) cost of fencing at ₹15 per metre.

Solution: (a) Area = length × breadth = 50 × 30 = 1500 m² (b) Perimeter = 2(50 + 30) = 2 × 80 = 160 m (c) Cost = 160 × 15 = ₹2400

Example 2: Circle and Cost Problem A circular garden has a radius of 14 m. Find the area to be turfed and the cost at ₹10 per m². Use π = 22/7.

Solution: Area = πr² = (22/7) × 14 × 14 = (22/7) × 196 = 22 × 28 = 616 m² Cost = 616 × 10 = ₹6160

Example 3: Cylinder Volume (Water Tank) A cylindrical water tank has radius 1.4 m and height 5 m. How many litres of water can it hold? (1 m³ = 1000 litres, π = 22/7)

Solution: Volume = πr²h = (22/7) × 1.4 × 1.4 × 5 = (22/7) × 1.96 × 5 = 22 × 0.28 × 5 = 22 × 1.4 = 30.8 m³ Capacity in litres = 30.8 × 1000 = 30,800 litres

Example 4: Cone Surface Area A cone has base radius 7 cm and slant height 25 cm. Find its curved surface area. (π = 22/7)

Solution: Curved Surface Area = πrl = (22/7) × 7 × 25 = 22 × 25 = 550 cm²

Common Mistakes

• Confusing radius and diameter: The question gives diameter 10 cm, but you use 10 as radius in πr². Always halve the diameter to get radius.
• Mixing up perimeter and area formulas: For a rectangle, writing Area = 2(l + b) instead of Perimeter. Double-check which quantity the question asks for.
• Forgetting unit conversions: Calculating area in metres when dimensions are in centimetres, or vice versa. Convert all measurements to the same unit before applying formulas.
• Using wrong π value: Some problems specify π = 22/7, others say 3.14. Read carefully. If nothing is mentioned, default to 22/7 for SSC GD unless the numbers favor 3.14.
• Lateral vs. total surface area confusion: The problem says "paint four walls" but you include the ceiling and floor in your calculation. Lateral surface area excludes top and bottom faces.
• Not simplifying fractions: Leaving an answer as (22/7) × 49 instead of calculating 22 × 7 = 154. SSC GD expects numerical answers, not unsimplified expressions.

Quick Reference

• Rectangle: A = l×b, P = 2(l+b); Square: A = a², P = 4a
• Circle: A = πr², C = 2πr; Semicircle: A = πr²/2, P = πr + 2r
• Triangle: A = ½×base×height; Equilateral: A = (√3/4)a²
• Cube: V = a³, TSA = 6a²; Cuboid: V = l×b×h, TSA = 2(lb+bh+hl)
• Cylinder: V = πr²h, CSA = 2πrh; Cone: V = (1/3)πr²h, CSA = πrl
• Sphere: V = (4/3)πr³, SA = 4πr²; Hemisphere: V = (2/3)πr³, CSA = 2πr²
• Always convert to common units; 1 m³ = 1000 litres; check if radius or diameter is given