Mensuration Quick Reference - Quick Revision
Mensuration Quick Reference - Quick Revision
Key Points (One-Liners)
- Perimeter is the total length of the boundary; Area is the space inside.
- Cube has 6 equal squares; its space diagonal is a√3.
- Cylinder curved surface = 2πrh; Total surface = 2πr(r + h).
- Sphere volume = (4/3)πr³; Surface = 4πr² (no pi-r-squared!).
- Cone slant height l = √(h² + r²); Volume = (1/3)πr²h.
- Prism volume = Base Area × Height; Pyramid volume = ⅓ × Base Area × Height.
- For frustum, subtract the small cone from the big cone.
- 1 hectare = 10,000 m²; 1 acre ≈ 4047 m².
- Circle area in terms of diameter: πD²/4.
- Diagonal of cuboid = √(l² + b² + h²).
- Ratio of volumes of similar solids = cube of the ratio of corresponding sides.
- Thickness of hollow cylinder = (R – r); volume of material = πh(R² – r²).
- Area of path around rectangular garden = 2w(l + b + 2w).
- Equilateral triangle height = (√3/2) × side.
- Maximum area for given perimeter is always the circle.
| Formula/Rule |
Application |
| Area of trapezium = ½ × (sum of |
|
| Volume of hollow sphere = (4/3)π(R³ – r³) |
Ball bearings, metal shells |
| Surface area of hemisphere = 3πr² |
Dome painting, half-water tanks |
| Length of wire drawn from melted sphere = ( sphere volume ) / ( πr² wire ) |
Wire-drawing problems |
| Diagonal of square = a√2 |
Tiles fitting diagonally |
| Area of sector = (θ/360) × πr² |
Pizza/gear slice problems |
| Volume of cap = (1/3)πh²(3R – h) |
Spherical tank ends |
| Area of four walls = 2h(l + b) |
Room painting (no floor/ceiling) |
Memory Tricks
- “CCC” – Cube: Curve surface nil, Constant area 6a², Capacity a³.
- “Two-pies” day – Anything rolled (cylinder, cone) has 2π in curved surface.
- “Volume thirds” – Cone, Pyramid, Frustum → all carry ⅓.
- “Sphere surface 4, volume 4/3” – 4 is the magic number.
- “LBH” – Length Breadth Height always multiply for cuboid volume.
Common Mistakes
| Mistake |
Correct Approach |
| Using πr²h for cone volume |
Remember ⅓πr²h |
| Forgetting to add top & bottom in cylinder total surface |
Use 2πr² + 2πrh |
| Taking slant height = height in cone |
Use l = √(h² + r²) |
| Calculating area of path outside only once |
Use outer – inner or 2w(l + b + 2w) |
| Mixing diameter with radius in formulas |
Always halve the given diameter first |
Last Minute Tips
- Write all formulas on a single flash card; glance right before exam starts.
- Mark units in every step—m², cm³—avoids silly conversion traps.
- Draw tiny sketch even for 1-marker; prevents radius-height swap.
- Approximate π as 22/7 unless question says 3.14; saves calculation time.
- Do dimension check: volume must be length³, area length²—catches blunders.
Quick Practice (5 MCQs)
1. The volume of a sphere is 4851 cm³. Find its radius. (Take π = 22/7)
4851 = 4/3 × 22/7 × r³ ⇒ r³ = 9261/8 ⇒ r = 10.5 cm
Ans: 10.5 cm
2. A 14 m × 10 m rectangular park has a 2 m wide path inside. Area of path?
Outer area = 140 m²; inner = (14–4)(10–4) = 60 m²; path = 140 – 60 = 80 m²
Ans: 80 m²
3. Curved surface of a cone (r = 7 cm, h = 24 cm) is:
l = √(7²+24²)=25 cm; CSA = πrl = 22/7×7×25 = 550 cm²
Ans: 550 cm²
4. How many 6 cm cubes fit into a 60 cm × 48 cm × 36 cm box?
Along edges: 10 × 8 × 6 = 480
Ans: 480
5. A hemispherical bowl of internal radius 9 cm is full of water. The water is poured into 27 identical cylindrical bottles each of radius 3 cm. Find height of water in each bottle.
Volume of water = 2/3 π(9)³ = 486π cm³; each bottle gets 486π/27 = 18π = π(3)²h ⇒ h = 2 cm
Ans: 2 cm
BETA
