Mensuration

Overview

Mensuration is the branch of mathematics dealing with the measurement of geometric figures—their lengths, areas, and volumes. For AP TET Paper I (Classes 1-5) and Paper II (Classes 6-8), this topic forms a substantial portion of the mathematics section, typically contributing 3-5 questions per paper.

At the primary level, students encounter perimeter and area of basic 2D shapes (rectangles, squares, triangles, circles). At the upper primary level, the scope expands to include surface area and volume of 3D solids (cubes, cuboids, cylinders, cones, spheres). Mastery requires memorising formulas and—equally important—understanding when and how to apply them in word problems.

Questions often combine mensuration with real-life contexts: finding the cost of fencing a field, the amount of paint needed for walls, or the capacity of a tank. Expect both direct formula application and multi-step reasoning problems.

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Key Concepts

• **Perimeter** is the total length of the boundary of a 2D shape. Think of it as the distance you would walk if you traced the outline.

• **Area** measures the surface enclosed within a 2D boundary, expressed in square units (cm², m²).

• **Surface Area** of a 3D solid is the total area of all its outer faces. Distinguish between **Curved Surface Area (CSA)**—the curved part only—and **Total Surface Area (TSA)**—curved plus flat faces.

• **Volume** measures the space occupied by a 3D object, expressed in cubic units (cm³, m³). For liquids, 1000 cm³ = 1 litre.

• **Unit Consistency**: Always convert measurements to the same unit before calculating. Common conversions: 1 m = 100 cm; 1 km = 1000 m; 1 m² = 10,000 cm².

• **Composite Figures**: Many problems involve shapes made by combining or removing standard figures. Break them into simpler parts, calculate separately, then add or subtract.

• **Practical Applications**: Fencing (perimeter), tiling/painting (area), constructing tanks/boxes (surface area/volume), filling containers (volume in litres).

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Formulas / Key Facts

### 2D Figures — Perimeter and Area

| Figure | Perimeter | Area | |--------|-----------|------| | Square (side a) | 4a | a² | | Rectangle (length l, breadth b) | 2(l + b) | l × b | | Triangle (sides a, b, c; base b, height h) | a + b + c | ½ × b × h | | Right Triangle (legs a, b) | a + b + √(a² + b²) | ½ × a × b | | Equilateral Triangle (side a) | 3a | (√3/4) × a² | | Parallelogram (base b, height h, side a) | 2(a + b) | b × h | | Rhombus (diagonals d₁, d₂; side a) | 4a | ½ × d₁ × d₂ | | Trapezium (parallel sides a, b; height h) | Sum of all sides | ½ × (a + b) × h | | Circle (radius r) | 2πr (circumference) | πr² | | Semicircle (radius r) | πr + 2r | ½ × πr² |