Mensuration

Area, Surface Area and Volume of 2D/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 TN TET Paper II, this topic carries significant weight as it tests both conceptual understanding and computational accuracy. Questions typically involve direct formula application, word problems requiring identification of appropriate shapes, and comparison problems.

Students must master two broad categories: **2D figures** (area and perimeter) and **3D figures** (surface area and volume). The syllabus expects familiarity with standard shapes taught in classes 6-8: rectangles, triangles, circles, parallelograms, trapeziums, cubes, cuboids, cylinders, cones and spheres. Real-life application questions — finding the cost of painting a wall, volume of a water tank, material needed for a box — are common.

Success requires memorising formulas accurately, understanding when each applies, and careful unit conversion. Careless errors in squaring, cubing or forgetting to convert cm to m are frequent pitfalls.

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Key Concepts

• **Perimeter** is the total length of the boundary of a 2D figure; **area** is the region enclosed within that boundary.

• **Surface area** of a 3D object is the total area of all its outer faces. It comes in two types: **Curved/Lateral Surface Area (CSA/LSA)** covers only the curved or side surfaces; **Total Surface Area (TSA)** includes top and bottom as well.

• **Volume** measures the space occupied by a 3D object — think of it as the capacity to hold water or sand.

• For composite figures, break them into standard shapes, calculate separately, then add or subtract as needed.

• Units matter: area is in square units (cm², m²), volume is in cubic units (cm³, m³). 1 m² = 10,000 cm²; 1 m³ = 1,000,000 cm³.

• **π (pi)** is taken as 22/7 or 3.14 unless specified otherwise in the question.

• A **hemisphere** is half a sphere — its CSA excludes the flat circular base; TSA includes it.

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Formulas / Key Facts

### 2D Figures

| Figure | Perimeter | Area | |--------|-----------|------| | Rectangle (l × b) | 2(l + b) | l × b | | Square (side a) | 4a | a² | | Triangle (base b, height h) | sum of sides | (1/2) × b × h | | Equilateral Triangle (side a) | 3a | (√3/4) × a² | | Circle (radius r) | 2πr (circumference) | πr² | | Semicircle | πr + 2r | (1/2)πr² | | Parallelogram (base b, height h) | 2(a + b) | b × h | | Rhombus (diagonals d₁, d₂) | 4 × side | (1/2) × d₁ × d₂ | | Trapezium (parallel sides a, b; height h) | sum of sides | (1/2) × (a + b) × h |

### 3D Figures

| Figure | CSA/LSA | 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) | (1/3)πr²h | | Sphere (r) | 4πr² | 4πr² | (4/3)πr³ | | Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |

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

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Worked Examples

### Example 1: Area of Trapezium *Find the area of a trapezium with parallel sides 12 cm and 8 cm, and height 5 cm.*

**Solution:** Area = (1/2) × (sum of parallel sides) × height Area = (1/2) × (12 + 8) × 5 Area = (1/2) × 20 × 5 = 50 cm²

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

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

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

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### Example 3: Composite Figure *A solid consists of a cone placed on top of a hemisphere. Both have radius 3 cm. The height of the cone is 4 cm. Find the total surface area. (Take π = 22/7)*

**Solution:** First, find slant height of cone: l = √(r² + h²) = √(9 + 16) = √25 = 5 cm

CSA of cone = πrl = (22/7) × 3 × 5 = 330/7 cm²

CSA of hemisphere = 2πr² = 2 × (22/7) × 9 = 396/7 cm²

Total surface area = 330/7 + 396/7 = 726/7 ≈ 103.7 cm²

*(Note: The flat circular faces of cone and hemisphere are joined, so not counted.)*

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Common Mistakes

• **Confusing radius and diameter** → Always check: if diameter is given, halve it to get radius before applying formulas.

• **Using area formula for perimeter or vice versa** → Read the question carefully. "Fencing" or "boundary" means perimeter; "painting a floor" or "tiling" means area.

• **Forgetting to square or cube** → Area involves r² or a²; volume involves r³ or a³. Write the formula first, then substitute.

• **Mixing up CSA and TSA** → CSA excludes top/bottom; TSA includes them. "Painting the curved surface only" needs CSA; "total material for making a closed box" needs TSA.

• **Ignoring units or wrong conversion** → If length is in cm and answer required in m², convert first. Remember: 1 m = 100 cm, so 1 m² = 10,000 cm².

• **Using wrong value of π** → Use the value specified in the question (22/7 or 3.14). If not specified, 22/7 is standard for TN TET.

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Quick Reference

• **Rectangle area = l × b; Perimeter = 2(l + b)**
• **Circle area = πr²; Circumference = 2πr**
• **Trapezium area = (1/2)(a + b) × h**
• **Cube: TSA = 6a², Volume = a³**
• **Cylinder: Volume = πr²h, TSA = 2πr(r + h)**
• **Cone volume = (1/3)πr²h; Slant height l = √(r² + h²)**
• **Sphere volume = (4/3)πr³; Surface area = 4πr²**