Mensuration (Class 6–8)

Overview

Mensuration is the branch of mathematics dealing with the measurement of geometric figures—their lengths, areas, and volumes. For UPTET Paper II, this topic bridges arithmetic skills with spatial reasoning, testing whether candidates can apply formulas to everyday objects like fields, tanks, rooms, and containers. Questions typically involve calculating perimeter and area of plane figures (rectangles, triangles, circles, parallelograms, trapeziums) and surface area and volume of solids (cubes, cuboids, cylinders, cones, spheres).

This is a high-scoring section because formulas, once memorised, can be directly applied. However, examiners often set traps by mixing units (cm and m), giving diameter instead of radius, or asking for total vs curved surface area. Mastery here requires knowing which formula applies to which shape, careful unit conversion, and quick mental checks to verify reasonableness of answers.

Key Concepts

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

• **Surface area** of a 3D solid is the total area of all its outer faces; it can be **curved surface area (CSA)** (excluding flat ends) or **total surface area (TSA)** (including all faces).

• **Volume** is the amount of 3D space a solid occupies, measured in cubic units (cm³, m³, litres where 1 litre = 1000 cm³).

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

• Unit consistency is critical: convert all measurements to the same unit before applying any formula.

• The value of π is taken as 22/7 or 3.14 unless otherwise specified; use whichever simplifies calculation.

• Doubling linear dimensions quadruples area and multiplies volume by eight—useful for ratio-based questions.

Formulas / Key Facts

### 2D Figures — Perimeter (P) and Area (A)

| 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 | | 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 |

### 3D Figures — Surface Area and Volume

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