Mensuration
Area, Surface Area and Volume of Solids
Overview
Mensuration is the branch of mathematics dealing with measurement of geometric figures—their lengths, areas and volumes. For Assam TET Paper II, this topic carries significant weightage as it tests both conceptual understanding and computational accuracy. Questions typically involve calculating areas of plane figures, surface areas of solids, and volumes of three-dimensional objects.
Mastery of mensuration requires memorizing key formulas and understanding when to apply each. The topic connects directly to real-life applications—calculating land area, capacity of tanks, material needed for construction—making it relevant for teaching upper primary students.
Key Concepts
- Area measures the surface enclosed by a plane figure, expressed in square units (cm², m²).
- Perimeter is the total length of the boundary of a plane figure, expressed in linear units (cm, m).
- Surface area of a solid is the total area of all its outer faces—distinguish between curved surface area (CSA) and total surface area (TSA).
- Volume measures the space occupied by a solid, expressed in cubic units (cm³, m³); 1 litre = 1000 cm³.
- Lateral surface area refers to the area of the sides only, excluding top and bottom bases.
- Right solids have their axis perpendicular to the base—all TET formulas assume right solids unless stated otherwise.
- Composite figures require breaking down into simpler shapes, calculating separately, then adding or subtracting as needed.
Formulas / Key Facts
Plane Figures (Area and Perimeter)
| Figure | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Triangle | (1/2) × base × height | Sum of three sides |
| Equilateral Triangle | (√3/4) × a² | 3a |
| Parallelogram | base × height | 2(a + b) |
| Rhombus | (1/2) × d₁ × d₂ | 4a |
| Trapezium | (1/2) × (a + b) × h | Sum of all sides |
| Circle | πr² | 2πr (circumference) |
| Semicircle | (1/2)πr² | πr + 2r |
Three-Dimensional Solids
| Solid | CSA / LSA | TSA | Volume |
|---|---|---|---|
| Cube (side 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, l) | πrl | πr(r + l) | (1/3)πr²h |
| Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |
Note: For cone, slant height l = √(r² + h²)
Important Conversions
- 1 m = 100 cm; 1 m² = 10000 cm²; 1 m³ = 1000000 cm³
- 1 litre = 1000 cm³ = 0.001 m³
- 1 hectare = 10000 m²
Worked Examples
Example 1: Volume and Surface Area of Cylinder
Problem: A cylindrical water tank has radius 7 m and height 10 m. Find its volume and total surface area. (Use π = 22/7)
Solution:
- Volume = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 70 = 1540 m³
- TSA = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 2 × 22 × 17 = 748 m²
Example 2: Cone Calculations
Problem: A cone has radius 6 cm and height 8 cm. Find the slant height, curved surface area and volume. (Use π = 3.14)
Solution:
- Slant height l = √(r² + h²) = √(36 + 64) = √100 = 10 cm
- CSA = πrl = 3.14 × 6 × 10 = 188.4 cm²
- Volume = (1/3)πr²h = (1/3) × 3.14 × 36 × 8 = (1/3) × 904.32 = 301.44 cm³
Example 3: Composite Solid
Problem: A solid consists of a hemisphere mounted on a cylinder. Both have radius 3 cm and the cylinder height is 5 cm. Find the total surface area.
Solution:
- CSA of cylinder = 2πrh = 2 × π × 3 × 5 = 30π cm²
- CSA of hemisphere = 2πr² = 2 × π × 9 = 18π cm²
- Base area of cylinder = πr² = 9π cm² (only bottom, top is covered by hemisphere)
- TSA = 30π + 18π + 9π = 57π = 179.07 cm² (using π = 3.14)
Common Mistakes
- Confusing CSA with TSA → CSA excludes bases; TSA includes all surfaces. Read the question carefully for keywords "curved," "lateral," or "total."
- Forgetting to calculate slant height for cone → Students directly use height in CSA formula. Always compute l = √(r² + h²) first when slant height is not given.
- Unit inconsistency → Mixing cm and m in the same calculation leads to wrong answers. Convert all measurements to the same unit before applying formulas.
- Using diameter instead of radius → Formulas use radius. When diameter is given, always halve it before substitution.
- Wrong fraction in volume formulas → Cone uses (1/3)πr²h, hemisphere uses (2/3)πr³, sphere uses (4/3)πr³. Memorize these fractions distinctly.
- Ignoring the open top/bottom → For open tanks or hollow cylinders, exclude the open face from TSA calculation.
Quick Reference
- Cube: TSA = 6a², Volume = a³
- Cylinder: Volume = πr²h, TSA = 2πr(r + h)
- Cone: Volume = (1/3)πr²h, Slant height = √(r² + h²)
- Sphere: Volume = (4/3)πr³, Surface area = 4πr²
- Hemisphere: Volume = (2/3)πr³, TSA = 3πr²
- 1 litre = 1000 cm³ — essential for capacity problems