Mensuration — Study Notes for UP Police Constable

Overview

Mensuration is the branch of mathematics dealing with measurement of geometric figures — their lengths, areas, surface areas and volumes. In UP Police Constable exam, this topic typically yields 3–5 questions and tests your ability to apply standard formulas to practical scenarios like finding costs of painting walls, carpeting floors, filling tanks, or comparing quantities.

Success in mensuration requires two skills: accurate recall of formulas and careful attention to units. The exam favours straightforward problems — you're rarely asked to derive formulas or solve complex multi-step geometry. Instead, questions test whether you can identify the correct shape, plug values into the right formula, and perform arithmetic correctly. Common question types include finding area given dimensions, calculating volume of containers, determining perimeter for fencing problems, and cost-based word problems that multiply area/volume by rate per unit.

Master the formulas for rectangles, squares, circles, triangles, cubes, cuboids, cylinders, cones and spheres. Practice unit conversions (m to cm, litre to m³) and recognize that careless calculation errors cost more marks in mensuration than lack of conceptual understanding.

Key Concepts

• **2-D vs 3-D figures**: Two-dimensional shapes (plane figures) have area and perimeter; three-dimensional solids have surface area and volume. Don't confuse them in exam pressure.

• **Perimeter is boundary length**: The distance around a closed figure. Used in fencing, framing and border problems. Measured in linear units (m, cm, km).

• **Area is surface coverage**: The space enclosed within a 2-D boundary. Used in flooring, painting flat surfaces and land measurement. Measured in square units (m², cm², hectare).

• **Surface area for solids**: Total area of all outer surfaces of a 3-D object. Lateral (curved) surface area excludes top/bottom. Total surface area includes all faces. Used in painting, wrapping and material estimation.

• **Volume is space occupied**: The three-dimensional capacity of a solid. Used in filling, emptying, capacity and displacement problems. Measured in cubic units (m³, cm³, litre where 1 litre = 1000 cm³).

• **Unit consistency is critical**: Always convert all measurements to the same unit before applying formulas. Mixing metres and centimetres causes wrong answers.

• **Cost problems multiply quantity by rate**: If area = 50 m² and cost = ₹20/m², total cost = 50 × 20 = ₹1000. Watch for unit rates carefully.

Formulas / Key Facts

**Two-Dimensional Figures:**

• **Rectangle**: Perimeter = 2(length + breadth); Area = length × breadth
• **Square**: Perimeter = 4 × side; Area = side²
• **Triangle**: Perimeter = sum of three sides; Area = ½ × base × height. For equilateral triangle with side a: Area = (√3/4)a²
• **Circle**: Circumference = 2πr = πd (where r = radius, d = diameter); Area = πr²
• **Semicircle**: Perimeter = πr + 2r; Area = πr²/2
• **Trapezium**: Area = ½ × (sum of parallel sides) × height
• **Parallelogram**: Area = base × height
• **Rhombus**: Area = ½ × product of diagonals

**Three-Dimensional Figures:**

• **Cube** (all edges equal, side = a): Total Surface Area = 6a²; Volume = a³
• **Cuboid** (length l, breadth b, height h): Total Surface Area = 2(lb + bh + hl); Volume = l × b × h
• **Cylinder** (radius r, height h): Curved Surface Area = 2πrh; Total Surface Area = 2πr(h + r); Volume = πr²h
• **Cone** (radius r, height h, slant height l where l² = r² + h²): Curved Surface Area = πrl; Total Surface Area = πr(l + r); Volume = (1/3)πr²h
• **Sphere** (radius r): Surface Area = 4πr²; Volume = (4/3)πr³
• **Hemisphere** (radius r): Curved Surface Area = 2πr²; Total Surface Area = 3πr²; Volume = (2/3)πr³

**Common Unit Conversions:**

• 1 m = 100 cm; 1 km = 1000 m
• 1 m² = 10000 cm²; 1 hectare = 10000 m²
• 1 m³ = 1000000 cm³; 1 litre = 1000 cm³
• Use π = 22/7 or 3.14 as specified in question

Worked Examples

**Example 1 — Area and Cost (Rectangle)**

A rectangular hall is 15 m long and 8 m wide. Find the cost of carpeting it at ₹125 per m².

*Solution:*

• Area of rectangular floor = length × breadth = 15 × 8 = 120 m²
• Cost = Area × Rate = 120 × 125 = ₹15,000

**Example 2 — Volume (Cylinder)**

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)² × 2
• = (22/7) × 1.96 × 2 = (22/7) × 3.92 = 22 × 0.56 = 12.32 m³
• Convert to litres: 1 m³ = 1000 litres
• Capacity = 12.32 × 1000 = 12,320 litres

**Example 3 — Surface Area (Cube)**

The edge of a cube is 5 cm. Find its total surface area and volume.

*Solution:*

• Side a = 5 cm
• Total Surface Area = 6a² = 6 × (5)² = 6 × 25 = 150 cm²
• Volume = a³ = (5)³ = 125 cm³

**Example 4 — Combination (Cone)**

Find the curved surface area of a cone with base radius 7 cm and height 24 cm. (π = 22/7)

*Solution:*

• First find slant height: l² = r² + h² = 7² + 24² = 49 + 576 = 625; so l = 25 cm
• Curved Surface Area = πrl = (22/7) × 7 × 25 = 22 × 25 = 550 cm²

Common Mistakes

• **Using perimeter formula for area**: Students often confuse 2(l+b) with l×b. Remember: perimeter is boundary, area is surface. *Fix: Read question carefully — "fencing" means perimeter, "flooring/painting" means area.*

• **Forgetting to convert units**: Mixing metres and centimetres in the same calculation produces wrong results. *Fix: Convert all measurements to the same unit before applying any formula.*

• **Confusing curved and total surface area**: For cylinders/cones, curved surface excludes circular bases; total surface includes them. *Fix: "Curved/lateral" = sides only; "Total" = all surfaces including top/bottom.*

• **Wrong volume-to-litre conversion**: 1 litre = 1000 cm³, not 1000 m³. Also 1 m³ = 1000 litres. *Fix: Remember 1 m³ = 10⁶ cm³ = 1000 litres. Convert carefully.*

• **Squaring/cubing errors**: (0.5)² ≠ 1, it equals 0.25. Students rush and miscalculate. *Fix: Write intermediate steps, especially for decimals and fractions.*

• **Misidentifying the shape**: A water tank on its side might be a cylinder with different orientation. *Fix: Visualize or sketch the figure before selecting formula.*

Quick Reference

• **Rectangle**: Area = l×b; Perimeter = 2(l+b)
• **Square**: Area = a²; Perimeter = 4a
• **Circle**: Area = πr²; Circumference = 2πr
• **Triangle**: Area = ½ × base × height
• **Cube**: Volume = a³; Surface Area = 6a²
• **Cuboid**: Volume = l×b×h; Surface Area = 2(lb+bh+hl)
• **Cylinder**: Volume = πr²h; Curved SA = 2πrh; Total SA = 2πr(h+r)
• **Cone**: Volume = (1/3)πr²h; Curved SA = πrl
• **Sphere**: Volume = (4/3)πr³; Surface Area = 4πr²
• **Hemisphere**: Volume = (2/3)πr³; Curved SA = 2πr²; Total SA = 3πr²
• **1 m³ = 1000 litres = 10⁶ cm³; 1 hectare = 10000 m²**