Dosto, ab aate hain 3D (solid) aakritiyon par — cube (ghan), cuboid (ghanabh), cylinder (belan), cone (shanku) aur sphere (gola). Do cheezein poochi jaati hain:
Volume (Aayatan) = andar kitna maal/paani samayega (unit cm³, m³, litre) · Surface Area (Prishth kshetrafal) = bahar ki poori satah (unit cm², m²) — jaise dabbe par rang karna ya kagaz chipkana.
CET Group D mein 1–2 questions aate hain, aur ye seedhe formula wale hote hain. Chapter 11 ki tarah — formula yaad = marks pakke. Chalo table ratt lete hain! 💪
1. Cube (Ghan) aur Cuboid (Ghanabh)
| Kya | Cube (bhuja a) | Cuboid (l × b × h) |
|---|---|---|
| Volume | a³ | l × b × h |
| TSA (total surface) | 6a² | 2(lb + bh + hl) |
| LSA/CSA (4 deewarein) | 4a² | 2h(l + b) |
| Diagonal | a√3 | √(l² + b² + h²) |
TSA: 2(10×5 + 5×4 + 4×10) = 2(50 + 20 + 40) = 2 × 110 = 220 cm²
2. Cylinder (Belan) — ⭐ Sabse zyada aane wala
| Kya | Formula | Yaad rakhne ka tarika |
|---|---|---|
| Volume | πr²h | Circle ka area × unchai |
| CSA (curved) | 2πrh | Circumference × unchai (label/sticker) |
| TSA | 2πr(r + h) | CSA + do circle (2πr²) |
CSA: 2 × (22/7) × 7 × 10 = 440 cm²
TSA: 2 × (22/7) × 7 × (7+10) = 44 × 17 = 748 cm²
3. Cone (Shanku)
| Kya | Formula |
|---|---|
| Volume | ⅓ πr²h (cylinder ka teesra hissa!) |
| Slant height (l) | √(r² + h²) (Pythagoras) |
| CSA | πrl |
| TSA | πr(r + l) |
Slant height (agar r=3, h=4 ho) = √(9+16) = 5 cm (3-4-5 triplet!)
4. Sphere (Gola) aur Hemisphere (Ardh-gola)
| Kya | Sphere | Hemisphere |
|---|---|---|
| Volume | ⁴⁄₃ πr³ | ⅔ πr³ |
| CSA (curved) | 4πr² | 2πr² |
| TSA | 4πr² (wahi) | 3πr² (curved + flat circle) |
Volume (r = 3 ke liye): ⁴⁄₃ × π × 27 = 36π ≈ 113.1 cm³
5. Volume aur Litre ka Rishta
Dhyan: 1 m³ = 10,00,000 cm³ (10 lakh) aur 1 m² = 10,000 cm².
Step 1: Bhuja = ∛343 = 7 cm (cube root — 7³ = 343)
Step 2: TSA = 6 × 49 = 294 cm² ✓
Cube roots ratt lo: 27→3, 64→4, 125→5, 216→6, 343→7, 512→8, 729→9, 1000→10.
Bhuja teen guna → volume 27 guna, surface 9 guna.
Rule: volume par cube (³) ka asar, area par square (²) ka.
Ginti = bade ka volume ÷ chhote ka volume. Surface area badal jata hai, par volume kabhi nahi!
Step 1: Volume = 3 m³ → Step 2: × 1000 = 3000 litre ✓
Cone = ⅓ × cylinder
Same radius aur height ho to teen cone ka paani ek cylinder bharta hai. Isliye ⅓πr²h — alag se yaad karne ki zaroorat nahi!
CSA = 2πr × h
Bottle ke label ko kholo to rectangle banta hai — lambai = circumference (2πr), chaudai = height. Isliye CSA = 2πrh.
Sphere 4πr² · Hemisphere 2πr² (curved), 3πr² (total)
Hemisphere = aadha gola (2πr²) + neeche ka flat circle (πr²) = 3πr². Logic yaad rahe to formula bhi.
27·64·125·216·343·512·729
= 3, 4, 5, 6, 7, 8, 9. Ulta sawal ("volume 343 diya hai") mein turant bhuja mil jayegi.
Volume ³, Area ²
Maap k guna → volume k³ guna, area k² guna. Dugna karo to 8 guna aur 4 guna. Ye ek line poora Type-3 solve kar deti hai.
| Solid | Volume | Surface Area |
|---|---|---|
| Cube | a³ | TSA 6a² · LSA 4a² · diag a√3 |
| Cuboid | lbh | TSA 2(lb+bh+hl) · diag √(l²+b²+h²) |
| Cylinder | πr²h | CSA 2πrh · TSA 2πr(r+h) |
| Cone | ⅓πr²h | CSA πrl · TSA πr(r+l) · l = √(r²+h²) |
| Sphere | ⁴⁄₃πr³ | 4πr² |
| Hemisphere | ⅔πr³ | CSA 2πr² · TSA 3πr² |
| 1 litre | 1000 cm³ · 1 m³ = 1000 litre | |
| 1 m³ | 10,00,000 cm³ | |
| Maap k guna | Volume k³ guna · Area k² guna | |
| Cube roots | 27·64·125·216·343·512·729 → 3,4,5,6,7,8,9 | |
| Pighlana (melting) | Volume same rehta hai | |
Q1. The volume of a cube of edge 6 cm is:
Q2. The total surface area of a cube of edge 5 cm is:
Q3. The volume of a cuboid of dimensions 10 cm × 5 cm × 4 cm is:
Q4. The total surface area of a cuboid 10 cm × 5 cm × 4 cm is:
Q5. The volume of a cylinder of radius 7 cm and height 10 cm is (π = 22/7):
Q6. The curved surface area of a cylinder of radius 7 cm and height 10 cm is (π = 22/7):
Q7. The volume of a cone of radius 7 cm and height 6 cm is (π = 22/7):
Q8. The slant height of a cone whose radius is 3 cm and height 4 cm is:
Q9. The volume of a sphere of radius 3 cm is:
Q10. The surface area of a sphere of radius 7 cm is (π = 22/7):
Q11. The length of the diagonal of a cube of edge 4 cm is:
Q12. The volume of a hemisphere of radius 3 cm is:
Q13. The volume of a cube is 343 cm³. Its total surface area is:
Q14. If the edge of a cube is doubled, its volume becomes:
Volume = andar ki jagah (cm³), Surface area = bahar ki satah (cm²). Cube a³ / 6a², cuboid lbh / 2(lb+bh+hl), cylinder πr²h / CSA 2πrh, cone ⅓πr²h (cylinder ka teesra hissa!) jiska l = √(r²+h²), sphere ⁴⁄₃πr³ / 4πr², hemisphere ⅔πr³ / TSA 3πr². Do galtiyon se bacho: cone mein ⅓ bhoolna aur CSA-TSA mila dena. Unit yaad rakho: 1 litre = 1000 cm³, 1 m³ = 1000 litre. Aur maap k guna hone par volume k³ guna, area k² guna — bhuja dugni to volume 8 guna! 🚀