Mensuration

Area, Surface Area and Volume of Solids

---

Overview

Mensuration is the branch of mathematics dealing with measurement of geometric figures—their lengths, areas, and volumes. For JTET Paper II, this topic bridges arithmetic computation with spatial reasoning, testing both formula recall and application skills.

This topic carries significant weightage as questions appear in both the content section (direct calculations) and pedagogy section (teaching approaches for measurement concepts). Students must master two-dimensional figures (area, perimeter) and three-dimensional solids (surface area, volume). The ability to visualize shapes, select correct formulas, and handle unit conversions is essential.

Expect questions involving composite figures, real-life applications (cost of painting, capacity of tanks), and comparison problems. Strong command here also supports science topics like density, pressure, and fluid mechanics.

---

Key Concepts

• **Area** measures the surface enclosed by a 2D figure, expressed in square units (cm², m²). **Perimeter** is the total boundary length.

• **Surface Area** of a 3D solid is the total area of all its faces. **Lateral/Curved Surface Area (LSA/CSA)** excludes the base(s); **Total Surface Area (TSA)** includes all surfaces.

• **Volume** measures the space occupied by a 3D object, expressed in cubic units (cm³, m³). **Capacity** refers to the volume a hollow object can hold (often in litres).

• **Unit conversion is critical**: 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Also, 1 litre = 1000 cm³ = 0.001 m³.

• **Composite figures** require breaking shapes into standard components, calculating separately, then adding or subtracting as needed.

• **Slant height (l)** in cones and pyramids differs from vertical height (h). Use Pythagoras: l² = h² + r² for cones.

• **Hemisphere** is half a sphere. Its curved surface excludes the circular base; total surface includes it.

---

Formulas / Key Facts

### Two-Dimensional Figures

| Figure | Area | Perimeter | |--------|------|-----------| | Rectangle | l × b | 2(l + b) | | Square | a² | 4a | | Triangle | ½ × base × height | Sum of all sides | | Circle | πr² | 2πr (circumference) | | Parallelogram | base × height | 2(a + b) | | Trapezium | ½ × (sum of parallel sides) × height | Sum of all sides | | Rhombus | ½ × d₁ × d₂ | 4 × side |

### Three-Dimensional Solids

| Solid | Curved/Lateral SA | Total SA | Volume | |-------|-------------------|----------|--------| | Cuboid | 2h(l + b) | 2(lb + bh + hl) | l × b × h | | Cube | 4a² | 6a² | a³ | | Cylinder | 2πrh | 2πr(r + h) | πr²h | | Cone | πrl | πr(r + l) | ⅓πr²h | | Sphere | 4πr² | 4πr² | ⁴⁄₃πr³ | | Hemisphere | 2πr² | 3πr² | ⅔πr³ |

**Key facts:**

• For cone, slant height l = √(h² + r²)
• Diagonal of cuboid = √(l² + b² + h²)
• Diagonal of cube = a√3
• Use π = 22/7 or 3.14 as specified in the question

---

Worked Examples

### Example 1: Volume of a Cylinder **Problem:** Find the volume of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)

**Solution:**

• Volume = πr²h
• Volume = (22/7) × 7² × 10
• Volume = (22/7) × 49 × 10
• Volume = 22 × 7 × 10 = 1540 cm³

---

### Example 2: Total Surface Area of a Cone **Problem:** 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 = √(h² + r²) = √(64 + 36) = √100 = 10 cm
• TSA = πr(r + l) = 3.14 × 6 × (6 + 10)
• TSA = 3.14 × 6 × 16 = 301.44 cm²

---

### Example 3: Composite Figure (Practical Application) **Problem:** A room is 8 m long, 6 m wide, and 3 m high. Find the cost of painting its four walls at Rs 25 per m². The room has one door (2 m × 1 m) and two windows (1.5 m × 1 m each).

**Solution:**

• Lateral surface area of room = 2h(l + b) = 2 × 3 × (8 + 6) = 84 m²
• Area of door = 2 × 1 = 2 m²
• Area of two windows = 2 × (1.5 × 1) = 3 m²
• Area to be painted = 84 − 2 − 3 = 79 m²
• Cost = 79 × 25 = Rs 1975

---

Common Mistakes

• **Confusing radius with diameter** → Always check whether the question gives radius (r) or diameter (d). If diameter is given, divide by 2 before applying formulas.

• **Using height instead of slant height for cone's curved surface** → The formula πrl requires slant height. Calculate l = √(h² + r²) first if only height is given.

• **Forgetting to include both bases in TSA** → Cylinder TSA includes two circular bases; cone TSA includes one base. Read the question carefully—sometimes only CSA is asked.

• **Unit mismatch in calculations** → Convert all measurements to the same unit before calculating. A common error: mixing cm and m, leading to answers off by powers of 10 or 100.

• **Applying 2D formulas to 3D problems** → Area of a circle (πr²) is not the surface area of a sphere (4πr²). Visualize the shape before selecting the formula.

• **Arithmetic errors with fractions** → Volume of sphere (⁴⁄₃πr³) and hemisphere (⅔πr³) require careful fraction handling. Write intermediate steps to avoid mistakes.

---

Quick Reference

• **Cylinder volume** = πr²h; **Cone volume** = ⅓ of cylinder = ⅓πr²h
• **Sphere volume** = ⁴⁄₃πr³; **Hemisphere volume** = half of sphere = ⅔πr³
• **1 litre = 1000 cm³** — use this for capacity problems
• **Slant height of cone**: l = √(h² + r²) — never skip this step
• **TSA = CSA + Area of base(s)** — count bases correctly for each solid
• **For composite solids**: break into parts, calculate separately, then combine