Mensuration
Perimeter, Area, Surface Area and Volume of Solids
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Overview
Mensuration is the branch of mathematics dealing with measurement of geometric figures — their lengths, areas and volumes. For PSTET Paper II, this topic draws from Classes VI–VIII NCERT Mathematics and forms a reliable source of 2–4 questions. You must be comfortable with two-dimensional figures (perimeter and area) as well as three-dimensional solids (surface area and volume).
The topic tests both formula recall and application in word problems. Examiners often frame questions around real-life contexts — fencing a field, painting a room, filling a tank — so understanding what each formula measures is as important as memorising it. Mastery here also supports the pedagogy section, where you may be asked how to teach these concepts using concrete materials.
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Key Concepts
- **Perimeter** is the total length of the boundary of a 2-D figure. Units are linear (m, cm).
- **Area** is the measure of the surface enclosed by a 2-D figure. Units are square (m², cm²).
- **Surface Area** is the total area of all outer faces of a 3-D solid. Distinguish between Curved Surface Area (CSA) and Total Surface Area (TSA).
- **Volume** is the space occupied by a 3-D solid. Units are cubic (m³, cm³). 1 litre = 1000 cm³.
- **Unit conversion** is critical: 1 m = 100 cm, so 1 m² = 10000 cm² and 1 m³ = 10⁶ cm³.
- For composite figures, break into standard shapes, compute separately, then add or subtract as needed.
- Pedagogy tip: Use paper cut-outs for area and unit cubes/water displacement for volume to build conceptual understanding.
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Formulas / Key Facts
### 2-D Figures (Perimeter & Area)
| 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 | | Right Triangle (legs p, q) | p + q + hypotenuse | ½ × p × q | | Parallelogram (base b, height h) | 2(a + b) where a, b are adjacent sides | b × h | | Rhombus (diagonals d₁, d₂) | 4 × side | ½ × d₁ × d₂ | | Trapezium (parallel sides a, b; height h) | sum of all sides | ½ × (a + b) × h | | Circle (radius r) | Circumference = 2πr | πr² | | Semicircle | πr + 2r | ½ πr² |
*Use π = 22/7 or 3.14 as specified in the question.*
### 3-D Solids (Surface Area & Volume)
| Solid | Curved/Lateral SA | Total SA | Volume | |-------|-------------------|----------|--------| | Cuboid (l × b × h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h | | Cube (side a) | 4a² | 6a² | a³ | | 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 | (4/3)πr³ | | Hemisphere (radius r) | 2πr² | 3πr² | (2/3)πr³ |