Volume & Surface Area of Solids
Solids का Volume और Surface Area
Volume & Surface Area of Solids
- Mensuration
- Volume & Surface Area of Solids
Compute the volume and surface area of cubes, cuboids, cylinders, cones, spheres and hemispheres.
🎯 Learning Objective
Compute the volume and surface area of cubes, cuboids, cylinders, cones, spheres and hemispheres.
💡 Concept
- Cube: Volume = a³, TSA = 6a², LSA = 4a², diagonal = a√3
- Cuboid: Volume = l×b×h, TSA = 2(lb + bh + hl), diagonal = √(l² + b² + h²)
- Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r + h)
- Cone: Volume = ⅓πr²h, CSA = πrl, TSA = πr(r + l), slant l = √(r² + h²)
- Sphere: Volume = (4/3)πr³, Surface area = 4πr²
- Hemisphere: Volume = (2/3)πr³, CSA = 2πr², TSA = 3πr²
🧮 Key Formulas
Cube: V = a³, TSA = 6a²
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Cuboid: V = lbh, TSA = 2(lb+bh+hl)
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Cylinder: V = πr²h, CSA = 2πrh, TSA = 2πr(r+h)
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Cone: V = ⅓πr²h, CSA = πrl, l = √(r²+h²)
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Sphere: V = (4/3)πr³, SA = 4πr² ; Hemisphere TSA = 3πr²
✏️ Easy Example
Q. Find the volume and total surface area of a cube of side 5 cm.
- Volume = a³ = 5 × 5 × 5
- TSA = 6a² = 6 × 5 × 5 = 6 × 25
Answer: Volume = 125 cm³, TSA = 150 cm²
🇮🇳 Real-Life Example
A water tanker on a railway platform, a cement pillar, a cricket ball — every real object's capacity or paint needed is just a volume or surface-area sum.
📝 Exam-Level Example
Q. A cuboidal tank measures 8 m × 6 m × 5 m. Find its volume and total surface area.
- Volume = l × b × h = 8 × 6 × 5 = 240
- TSA = 2(lb + bh + hl) = 2(48 + 30 + 40)
- = 2 × 118
Answer: Volume = 240 m³, TSA = 236 m²
📝 Exam-Level Example
Q. Find the volume and curved surface area of a cylinder of radius 7 cm and height 10 cm (π = 22/7).
- Volume = πr²h = (22/7) × 7 × 7 × 10 = 22 × 7 × 10
- CSA = 2πrh = 2 × (22/7) × 7 × 10 = 2 × 22 × 10
Answer: Volume = 1540 cm³, CSA = 440 cm²
📝 Exam-Level Example
Q. A cone has radius 7 cm and height 24 cm. Find its slant height, volume and curved surface area (π = 22/7).
- Slant height l = √(r² + h²) = √(49 + 576) = √625 = 25
- Volume = ⅓πr²h = ⅓ × (22/7) × 49 × 24 = ⅓ × 22 × 7 × 24
- CSA = πrl = (22/7) × 7 × 25 = 22 × 25
Answer: Slant = 25 cm, Volume = 1232 cm³, CSA = 550 cm²
📝 Exam-Level Example
Q. Find the surface area of a sphere and the total surface area of a hemisphere, each of radius 7 cm (π = 22/7).
- Sphere SA = 4πr² = 4 × (22/7) × 49 = 4 × 22 × 7
- Hemisphere TSA = 3πr² = 3 × (22/7) × 49 = 3 × 22 × 7
Answer: Sphere = 616 cm², Hemisphere = 462 cm²
🪄 Memory Trick
A cone, a hemisphere and a cylinder of the same radius and height are in the volume ratio 1 : 2 : 3. Remember 'cone is one-third of its cylinder'.
⚠️ Common Mistakes
- ❌ Using height instead of slant height in cone CSA (CSA = πrl, not πrh)
- ❌ Taking hemisphere TSA as 2πr² (it is 3πr² — curved plus the flat base)
- ❌ Forgetting the ⅓ factor in the cone volume formula
🏆 Exam Tips
- ✅ Always find the cone's slant height first with l = √(r²+h²)
- ✅ Learn triples 3-4-5, 5-12-13, 7-24-25 to get slant heights instantly
📌 Summary
- Cube V = a³, TSA = 6a²; Cuboid V = lbh, TSA = 2(lb+bh+hl)
- Cylinder V = πr²h, CSA = 2πrh
- Cone V = ⅓πr²h, CSA = πrl, l = √(r²+h²)
- Sphere SA = 4πr²; Hemisphere TSA = 3πr²
- Volume ratio cone : hemisphere : cylinder = 1 : 2 : 3