Mensuration

Overview

Mensuration is the branch of mathematics dealing with the measurement of geometric figures—their lengths, areas, and volumes. For TS TET Paper I (Classes 1-5) and Paper II (Classes 6-8), this topic carries significant weight because it connects abstract formulas to real-world applications that students encounter daily: calculating floor area, fabric needed for curtains, or water capacity of a tank.

The exam tests two core competencies: (1) recall and correct application of standard formulas, and (2) the ability to solve word problems that require identifying the appropriate figure and formula. Paper I focuses on basic 2D shapes (rectangles, squares, triangles, circles) and simple perimeter/area calculations. Paper II extends to 3D solids—cubes, cuboids, cylinders, cones, and spheres—requiring surface area and volume computations.

Mastery here demands memorising formulas accurately, understanding when to use area versus perimeter versus volume, and being comfortable with unit conversions (cm to m, cm² to m², cm³ to litres).

Key Concepts

• **Perimeter** is the total length of the boundary of a 2D figure; measured in linear units (cm, m).

• **Area** is the measure of the surface enclosed by a 2D figure; measured in square units (cm², m²).

• **Surface Area** of a 3D solid is the total area of all its faces/surfaces; for curved solids, it includes curved surface area (CSA) and total surface area (TSA).

• **Volume** is the space occupied by a 3D solid; measured in cubic units (cm³, m³) or capacity units (litres, where 1 litre = 1000 cm³).

• **Distinction between CSA and TSA**: CSA excludes base/top areas; TSA includes all surfaces.

• **Unit consistency**: Always convert all measurements to the same unit before applying formulas.

• **Composite figures**: Break complex shapes into standard shapes, calculate separately, then add or subtract as needed.

Formulas / Key Facts

### 2D Figures — Perimeter and Area

| Figure | Perimeter | Area | |--------|-----------|------| | Rectangle (l × b) | 2(l + b) | l × b | | Square (side a) | 4a | a² | | Triangle (sides a, b, c; base b, height h) | a + b + c | ½ × b × h | | Equilateral Triangle (side a) | 3a | (√3/4) × a² | | Circle (radius r) | 2πr (circumference) | πr² | | Semicircle | πr + 2r | ½ × πr² | | Parallelogram (base b, height h) | 2(a + b) | b × h | | Rhombus (diagonals d₁, d₂) | 4 × side | ½ × d₁ × d₂ | | Trapezium (parallel sides a, b; height h) | sum of all sides | ½ × (a + b) × h |

### 3D Figures — Surface Area and Volume

| Solid | CSA | TSA | Volume | |-------|-----|-----|--------| | Cube (edge a) | 4a² | 6a² | a³ | | Cuboid (l × b × h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h | | Cylinder (r, h) | 2πrh | 2πr(r + h) | πr²h | | Cone (r, h, slant l) | πrl | πr(r + l) | ⅓ × πr²h | | Sphere (r) | — | 4πr² | ⁴⁄₃ × πr³ | | Hemisphere (r) | 2πr² | 3πr² | ⅔ × πr³ |

**Key fact**: Slant height of cone, l = √(r² + h²)

**Conversion**: 1 m³ = 1000 litres; 1 litre = 1000 cm³

Worked Examples

### Example 1: Area and Perimeter of Rectangle *A rectangular playground is 50 m long and 30 m wide. Find its perimeter and area.*

**Solution**:

• Perimeter = 2(l + b) = 2(50 + 30) = 2 × 80 = **160 m**
• Area = l × b = 50 × 30 = **1500 m²**

### Example 2: Volume of Cylinder *A cylindrical water tank has radius 7 m and height 10 m. Find its capacity in litres. (Use π = 22/7)*

**Solution**:

• Volume = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 70 = 1540 m³
• Convert to litres: 1540 × 1000 = **15,40,000 litres**

### Example 3: TSA of Cone *A cone has radius 6 cm and height 8 cm. Find its total surface area. (Use π = 3.14)*

**Solution**:

• First find slant height: l = √(r² + h²) = √(36 + 64) = √100 = 10 cm
• TSA = πr(r + l) = 3.14 × 6 × (6 + 10) = 3.14 × 6 × 16 = **301.44 cm²**

Common Mistakes

• **Confusing perimeter with area**: Perimeter is linear (metres), area is squared (square metres). Students add when they should multiply. → *Always check: perimeter = adding lengths; area = multiplying dimensions.*

• **Using diameter instead of radius**: Many problems give diameter; students forget to halve it. → *Read the problem twice; if given diameter d, use r = d/2.*

• **Forgetting to square or cube units**: When converting, 1 m = 100 cm, but 1 m² = 10,000 cm² and 1 m³ = 10,00,000 cm³. → *Convert measurements before calculating, not after.*

• **Mixing up CSA and TSA**: Questions may ask for "area to paint the curved surface only" (CSA) versus "total material needed" (TSA). → *Identify whether bases/tops are included.*

• **Wrong formula for cone/sphere volume**: Students often forget the fractional coefficient (⅓ for cone, ⁴⁄₃ for sphere). → *Memorise with mnemonic: "Cone is one-third cylinder; sphere is four-thirds of πr³."*

Quick Reference

• Rectangle area = l × b; perimeter = 2(l + b)
• Circle area = πr²; circumference = 2πr
• Cube volume = a³; TSA = 6a²
• Cylinder volume = πr²h; CSA = 2πrh
• Cone volume = ⅓πr²h; slant height l = √(r² + h²)
• Sphere volume = ⁴⁄₃πr³; surface area = 4πr²
• 1 m³ = 1000 litres; always convert units before solving