Mensuration
Overview
Mensuration is the branch of mathematics dealing with the measurement of geometric figures — their lengths, areas, and volumes.
The topic divides naturally into two parts: 2D figures (area and perimeter) and 3D figures (surface area and volume). Exam questions often combine multiple concepts — calculating the cost of painting a room, finding how many smaller objects fit into a larger container, or determining material needed for construction. Mastering the formulas is essential, but understanding when to apply each formula separates toppers from average performers.
Success in mensuration requires memorizing approximately 15–20 formulas and developing spatial visualization skills. Questions frequently involve unit conversions, so comfort with metric relationships (1 m = 100 cm, 1 m³ = 1000 litres) is equally important.
Key Concepts
- Perimeter is the total boundary length of a 2D figure; Area is the space enclosed within that boundary.
- Surface Area of a 3D object 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 3D object; capacity refers to the liquid it can hold (1 litre = 1000 cm³).
- For composite figures, break them into standard shapes, calculate separately, then add or subtract as needed.
- When a solid is melted and recast into another shape, the volume remains constant — this principle drives many exam problems.
- Scaling relationships: if dimensions are multiplied by k, area multiplies by k² and volume by k³.
- Always check units before calculation — convert everything to the same unit system first.
Formulas / Key Facts
2D Figures
| Figure | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a² |
| Rectangle (l × b) | 2(l + b) | l × b |
| Triangle (base b, height h) | Sum of three sides | ½ × b × h |
| Equilateral Triangle (side a) | 3a | (√3/4) × a² |
| Right Triangle (legs a, b) | a + b + √(a² + b²) | ½ × a × b |
| Scalene Triangle (sides a, b, c) | a + b + c | √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2 (Heron's formula) |
| Circle (radius r) | 2πr (circumference) | πr² |
| Semicircle (radius r) | πr + 2r | ½ × πr² |
| Ring/Annulus (radii R, r) | 2π(R + r) | π(R² − r²) |
| Parallelogram (base b, height h) | 2(a + b) | 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 |
3D Figures
| Figure | CSA | TSA | Volume |
|---|---|---|---|
| Cube (side 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, height h, slant l) | πrl | πr(r + l) | ⅓ × πr²h |
| Sphere (radius r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (radius r) | 2πr² | 3πr² | (2/3)πr³ |
Key relationships:
- Slant height of cone: l = √(r² + h²)
- Diagonal of cuboid: √(l² + b² + h²)
- Diagonal of cube: a√3
- Use π = 22/7 or 3.14 as specified in the question
Worked Examples
Example 1: Area of Composite Figure
Problem: A rectangular lawn 50 m × 40 m has a circular fountain of radius 7 m at its centre. Find the area of the grassy portion.
Solution:
- Area of rectangle = 50 × 40 = 2000 m²
- Area of circle = πr² = (22/7) × 7 × 7 = 154 m²
- Grassy area = 2000 − 154 = 1846 m²
Example 2: Volume and Recasting
Problem: A solid metallic sphere of radius 6 cm is melted and recast into a cone of height 24 cm. Find the radius of the cone.
Solution:
- Volume of sphere = (4/3)πr³ = (4/3) × π × 216 = 288π cm³
- Volume of cone = ⅓ × π × R² × h = ⅓ × π × R² × 24 = 8πR²
- Since volumes are equal: 288π = 8πR²
- R² = 36, so R = 6 cm
Example 3: Cost-Based Problem
Problem: The dimensions of a room are 12 m × 10 m × 4 m. If plastering costs ₹15 per m², find the cost of plastering the four walls.
Solution:
- Area of four walls = 2h(l + b) = 2 × 4 × (12 + 10) = 8 × 22 = 176 m²
- Cost = 176 × 15 = ₹2640
Common Mistakes
- Confusing CSA and TSA: Students use total surface area when only curved/lateral surface is asked (like painting only the curved part of a cylinder). → Always read whether the question asks for "curved," "lateral," or "total" surface area.
- Forgetting to square or cube when scaling: If radius doubles, area becomes 4 times (not 2 times) and volume becomes 8 times. → Remember: Area ∝ (dimension)², Volume ∝ (dimension)³.
- Unit conversion errors: Mixing cm and m in the same calculation gives absurd answers. → Convert all measurements to the same unit before substituting into formulas.
- Using diameter instead of radius: Many questions give diameter; students directly substitute without halving. → Circle the value and write "r = d/2" as your first step.
- Wrong formula for irregular quadrilaterals: Applying rectangle formula to parallelogram or rhombus. → For rhombus, use diagonal formula; for parallelogram, use base × height (not side × side).
- Neglecting thickness in hollow objects: For hollow cylinders or spheres, volume = outer volume − inner volume. → Identify inner and outer radii separately.
Quick Reference
- Square: Area = a², Perimeter = 4a
- Circle: Area = πr², Circumference = 2πr
- Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r+h)
- Sphere: Volume = (4/3)πr³, Surface Area = 4πr²
- Cone: Volume = ⅓πr²h, Slant height l = √(r²+h²)
- Recasting principle: Volume before = Volume after
- 1 m³ = 1000 litres = 10⁶ cm³