Mensuration

Area, Surface Area and Volume of 2D and 3D Figures

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Overview

Mensuration is the branch of mathematics dealing with measurement of geometric figures—their lengths, areas, and volumes. For KTET Category II/III, this topic bridges upper primary and high school mathematics, testing both computational accuracy and conceptual understanding of when to apply which formula.

Expect 3–5 questions combining direct formula application with word problems involving real-life contexts (water tanks, rooms to be painted, material costs). The examiners often test whether candidates can distinguish between perimeter and area, or between surface area and volume—concepts that students frequently confuse. Mastery here requires memorising key formulas and understanding the dimensional logic: perimeter is linear (units), area is square (units²), volume is cubic (units³).

Strong performance demands you visualise figures, identify what quantity is being asked, select the correct formula, and handle unit conversions cleanly.

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Key Concepts

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

• **Area** measures 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 outer faces—think of the material needed to wrap it.

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

• **Lateral Surface Area (LSA)** excludes the top and bottom faces; **Total Surface Area (TSA)** includes all faces.

• When a shape is **composite** (made of simpler shapes), break it into parts, calculate separately, then add or subtract as needed.

• Unit consistency is critical: convert all measurements to the same unit before substituting into formulas.

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Formulas / Key Facts

### 2D Figures

| Figure | Perimeter | Area | |--------|-----------|------| | Rectangle | 2(l + b) | l × b | | Square | 4a | a² | | Triangle | a + b + c | ½ × base × height | | Right Triangle | a + b + c | ½ × leg₁ × leg₂ | | Equilateral Triangle | 3a | (√3/4) × a² | | Parallelogram | 2(a + b) | base × height | | Rhombus | 4a | ½ × d₁ × d₂ | | Trapezium | sum of all sides | ½ × (a + b) × h | | Circle | 2πr (circumference) | πr² | | Semicircle | πr + 2r | ½ × πr² |

**Heron's Formula** for triangle with sides a, b, c: s = (a + b + c)/2 Area = √[s(s−a)(s−b)(s−c)]

### 3D Figures

| Solid | Lateral Surface Area | Total Surface Area | Volume | |-------|---------------------|-------------------|--------| | Cuboid | 2h(l + b) | 2(lb + bh + hl) | l × b × h | | Cube | 4a² | 6a² | a³ | | Cylinder | 2πrh | 2πr(r + h) | πr²h | | Cone | πrl (l = slant height) | πr(r + l) | ⅓πr²h | | Sphere | — | 4πr² | (4/3)πr³ | | Hemisphere | 2πr² | 3πr² | (2/3)πr³ |

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

**Diagonal of cuboid**: √(l² + b² + h²)

**Diagonal of cube**: a√3

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Worked Examples

### Example 1: Area of a Trapezium A trapezium has parallel sides 12 cm and 8 cm, and height 5 cm. Find its area.

**Solution:** Area = ½ × (sum of parallel sides) × height Area = ½ × (12 + 8) × 5 Area = ½ × 20 × 5 = 50 cm²

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### Example 2: Volume and Surface Area of a Cylinder A cylindrical water tank has radius 7 m and height 10 m. Find its volume and total surface area. (Use π = 22/7)

**Solution:** Volume = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 70 = 1540 m³

TSA = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 2 × 22 × 17 = 748 m²

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### Example 3: Composite Figure A room floor is rectangular (8 m × 6 m) with a semicircular alcove of diameter 6 m attached to one side. Find the total floor area. (Use π = 3.14)

**Solution:** Rectangle area = 8 × 6 = 48 m²

Semicircle: radius = 6/2 = 3 m Semicircle area = ½ × π × r² = ½ × 3.14 × 9 = 14.13 m²

Total area = 48 + 14.13 = 62.13 m²

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Common Mistakes

1. **Confusing radius and diameter** Wrong: Using diameter directly in πr²h. Correct: Always halve the diameter to get radius before substituting.

2. **Mixing up LSA and TSA** Wrong: Using lateral surface area when asked for total surface area (e.g., painting a closed box). Correct: Read the problem—"open tank" means exclude one face; "closed" means TSA.

3. **Forgetting slant height for cone** Wrong: Using vertical height in lateral surface area formula πrl. Correct: Calculate slant height l = √(r² + h²) first.

4. **Unit conversion errors** Wrong: Adding cm² and m² values directly. Correct: Convert to same unit first. Remember: 1 m² = 10000 cm².

5. **Using perimeter formula for area problems** Wrong: Calculating 2(l + b) when asked for floor area. Correct: Identify the dimension being asked—perimeter is boundary length, area is surface coverage.

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Quick Reference

• **Rectangle area**: l × b; **Perimeter**: 2(l + b)

• **Circle area**: πr²; **Circumference**: 2πr

• **Cylinder volume**: πr²h; **TSA**: 2πr(r + h)

• **Cone volume**: ⅓πr²h; **Slant height**: √(r² + h²)

• **Sphere volume**: (4/3)πr³; **Surface area**: 4πr²

• **1 litre = 1000 cm³ = 0.001 m³**