Mensuration
Overview
Mensuration deals with the measurement of geometric shapes — calculating perimeter, area, surface area, and volume. The questions typically involve basic 2D shapes (rectangles, circles, triangles) and simple 3D solids (cubes, cylinders, cuboids).
Mastery of mensuration requires memorizing key formulas and applying them quickly. Unlike other arithmetic topics, there's little room for shortcuts — you either know the formula or you don't. The good news: the formulas are finite and the application is straightforward. Focus on speed and accuracy rather than complex problem-solving strategies.
Most questions test direct formula application with one or two calculation steps. Occasionally, you'll encounter problems combining two shapes (like a square inscribed in a circle) or requiring you to find dimensions before calculating area/volume.
Key Concepts
- Perimeter is the total boundary length of a 2D shape; Area is the space enclosed within that boundary.
- Surface Area of a 3D solid is the total area of all its faces; Volume is the space it occupies.
- For 3D solids, distinguish between Curved Surface Area (CSA) — only the curved portion — and Total Surface Area (TSA) — curved plus flat faces.
- When a shape is inscribed in another (e.g., circle in square), their dimensions share a relationship — identify the connecting element (diagonal, diameter, side).
- Units matter: Area uses square units (cm², m²), Volume uses cubic units (cm³, m³). Converting between units requires squaring or cubing the conversion factor.
- A hemisphere is half a sphere — its CSA excludes the flat circular base, while TSA includes it.
- For composite shapes, break them into standard shapes, calculate separately, then add or subtract as needed.
Formulas / Key Facts
2D Shapes
| Shape | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a² |
| Rectangle (l × b) | 2(l + b) | l × b |
| Circle (radius r) | 2πr | πr² |
| Triangle (sides a, b, c) | a + b + c | ½ × base × height |
| Equilateral Triangle (side a) | 3a | (√3/4) × a² |
| Right Triangle (legs a, b) | a + b + √(a² + b²) | ½ × a × b |
| Parallelogram (base b, height h) | 2(a + b) | b × h |
| Trapezium (parallel sides a, b; height h) | a + b + c + d | ½ × (a + b) × h |
Key relationships:
- Diagonal of square = a√2
- Diagonal of rectangle = √(l² + b²)
- Circumference = π × diameter
3D Shapes
| Solid | Curved Surface Area | Total Surface Area | 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) | ⅓πr²h |
| Sphere (radius r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (radius r) | 2πr² | 3πr² | (2/3)πr³ |
Slant height of cone: l = √(r² + h²)
Diagonal of cuboid: √(l² + b² + h²)
Worked Examples
Example 1: Finding Area from Perimeter
The perimeter of a rectangular field is 80 m. If the length is 10 m more than the breadth, find the area.
Solution:
- Let breadth = b, then length = b + 10
- Perimeter: 2(l + b) = 80 → l + b = 40
- Substituting: (b + 10) + b = 40 → 2b = 30 → b = 15 m
- Length = 25 m
- Area = 25 × 15 = 375 m²
Example 2: Volume of a Cylinder
A cylindrical tank has diameter 14 m and height 5 m. Find its volume. (Use π = 22/7)
Solution:
- Diameter = 14 m → Radius = 7 m
- Volume = πr²h = (22/7) × 7 × 7 × 5
- Volume = 22 × 7 × 5 = 770 m³
Example 3: Composite Shape
A square of side 14 cm has a circle of maximum possible size cut from it. Find the area of the remaining portion. (π = 22/7)
Solution:
- Maximum circle that fits → diameter = side of square = 14 cm
- Radius = 7 cm
- Area of square = 14² = 196 cm²
- Area of circle = (22/7) × 7² = 154 cm²
- Remaining area = 196 − 154 = 42 cm²
Common Mistakes
- Confusing radius and diameter → Always check what's given. Diameter = 2 × radius. Half the radius when diameter is provided before applying formulas.
- Mixing up CSA and TSA → Read carefully whether the question asks for "curved surface" or "total surface." A closed cylinder's TSA includes two circular ends; CSA does not.
- Forgetting to square/cube when converting units → 1 m² = 10,000 cm² (not 100). 1 m³ = 1,000,000 cm³. Always square for area, cube for volume.
- Using wrong triangle area formula → For any triangle, area = ½ × base × height, where height is perpendicular to that base. For equilateral triangles only, use (√3/4)a².
- Miscalculating slant height vs vertical height in cones → Slant height (l) is the side length; vertical height (h) goes straight down. Related by l² = r² + h².
Quick Reference
- Square: Area = side², Diagonal = side × √2
- Circle: Area = πr², Circumference = 2πr
- Cylinder volume = πr²h; Cube volume = a³
- Sphere volume = (4/3)πr³; Hemisphere volume = (2/3)πr³
- TSA of cylinder = 2πr(r + h); CSA of cylinder = 2πrh
- Use π = 22/7 when radius/diameter is multiple of 7; else use 3.14