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 |