Surface Areas & Volumes | Class 10

🌟 Kyun Zaroori Hai?

🏭 Real Life Applications:

• Ice cream cone mein cone + hemisphere = volume of ice cream
• Water tank (cylinder) mein kitna paani? → Volume
• Gift box wrap karne ke liye paper kitna chahiye? → Surface area
• Candle jo cone hai, cylinder par rakhkha — material cost kya? → SA + Volume

📋 All Formulas — Quick Reference

🟫 Cuboid

l = length, b = breadth, h = height
TSA = 2(lb+bh+hl)
LSA = 2(l+b)h
V = lbh

🟦 Cube

Side = a
TSA = 6a²
LSA = 4a²
V = a³

🔴 Cylinder

r = radius, h = height
TSA = 2πr(r+h)
CSA = 2πrh
V = πr²h

🔺 Cone

r = base radius, h = height
l = slant = √(r²+h²)
TSA = πr(r+l)
CSA = πrl
V = (1/3)πr²h

⚽ Sphere

r = radius
TSA = 4πr²
V = (4/3)πr³

🔵 Hemisphere

r = radius
TSA = 3πr²
CSA = 2πr²
V = (2/3)πr³

🔗 Combined Solids — Surface Area

Key Rule: Combined solid ka TSA = visible surfaces ka sum ONLY.
Jahan do shapes join hote hain wahan ki surfaces count NAHI hoti (woh andar hain)!

Example 1: Capsule = Cylinder + 2 Hemispheres (medicine capsule)

TSA = CSA of cylinder + 2 × CSA of hemisphere
= 2πrh + 2 × (2πr²) = 2πr(h + 2r)
Volume = πr²h + 2×(2/3)πr³ = πr²h + (4/3)πr³ = πr²(h + 4r/3)

Example 2: Cone on top of cylinder (r=3, cone h=4, cylinder h=8). TSA aur Volume.

Slant of cone: l = √(9+16) = 5 cm
TSA = πrl (cone lateral) + 2πrh (cylinder lateral) + πr² (bottom circle)
Note: top circle of cylinder = base of cone → not counted in SA
TSA = π×3×5 + 2π×3×8 + π×9 = 15π + 48π + 9π = 72π ≈ 226.19 cm²

Volume = (1/3)πr²×4 + πr²×8 = (4/3)π×9 + 72π = 12π + 72π = 84π ≈ 263.89 cm³

Example 3: Cube ke upar ek pyramid (cube side=10, pyramid h=6). TSA nikalo.

Cube ka TSA = 6×100 = 600 cm². Lekin top face par pyramid hai → top face minus.
Pyramid slant height: l = √(5²+6²) = √61 cm
4 triangular faces of pyramid = 4 × (1/2 × 10 × √61) = 20√61 ≈ 156.2 cm²
TSA = 600 − 100 (top) + 156.2 = 656.2 cm²

♻️ Conversion of Solids

Jab ek solid ko pighal ke ya reshape karke dusra solid banate hain, to Volume preserved rahta hai.
Original volume = New shape ka volume

Example 4: Ek sphere (r=9) ko pighala ke chote spheres (r=3) banaye. Kitne banenge?

Volume of big sphere = (4/3)π×729 = 972π
Volume of small sphere = (4/3)π×27 = 36π
Number = 972π/36π = 27 spheres

Example 5: Lead cylinder (r=6cm, h=12cm) ko ek cone mein reshape kiya (r=6cm). Cone ki height?

Cylinder volume = π×36×12 = 432π cm³
Cone: (1/3)π×36×h = 432π → h = 432×3/36 = 36 cm

Example 6: Cylindrical tank (r=2m, h=4.5m) se conical vessel (r=0.5m, h=1.5m) mein paani bhara. Kitne vessels bharenge?

Cylinder V = π×4×4.5 = 18π m³
Cone V = (1/3)π×0.25×1.5 = 0.125π m³
Number = 18π/0.125π = 144 vessels

🌀 Frustum of a Cone

Frustum: Cone ke upar se ek chhota cone kata hua part (jaise truncated cone).
Example: Flower pot, bucket, lampshade!

Frustum Formulas (r₁ = small radius, r₂ = large radius)

Slant height: l = √[h² + (r₂−r₁)²]
CSA = π(r₁+r₂)l
TSA = π(r₁+r₂)l + π(r₁² + r₂²)
Volume = (πh/3)(r₁² + r₁r₂ + r₂²)

Example 7: Bucket (frustum): h=20cm, r₁=6cm, r₂=15cm. Volume aur CSA nikalo.

l = √(400 + 81) = √481 ≈ 21.93 cm
V = (π×20/3)(36 + 90 + 225) = (20π/3)(351) = 2340π ≈ 7350.9 cm³
CSA = π(6+15)(21.93) = π×21×21.93 ≈ 1447.5 cm²

📝 Practice Questions

Q1. Solid hemisphere ka radius 7 cm hai. TSA nikalo. (π=22/7)

TSA = 3πr² = 3×(22/7)×49 = 462 cm²

Q2. Ek cone ka radius 6 cm aur slant height 10 cm. TSA nikalo.

TSA = πr(r+l) = π×6×(6+10) = 96π ≈ 301.59 cm²

Q3. Ek cylinder (r=5, h=14) ke andar exactly ek sphere aur ek cone fit hote hain (same r). Sphere aur cone ki volume, cylinder ki volume se compare karo.

Cylinder V = π×25×14 = 350π. Sphere r=5: V=(4/3)π×125=500π/3. Cone r=5,h=14: V=(1/3)×25π×14=350π/3. Sum = (500+350)π/3=850π/3 ≈ 0.81×350π. Together about 81% of cylinder. Note: A sphere inscribed in cylinder: r=5, h=10 for sphere. This problem needs specific shape constraints.

Q4. Ek metallic cone (r=4.2, h=2.1) ko melt karke spheres (r=0.21cm) banao. Kitne spheres?

Cone V = (1/3)×π×4.2²×2.1 = (1/3)×π×18.522×2.1 = 12.965π cm³
Sphere V = (4/3)π×0.21³ = (4/3)π×0.009261 = 0.01235π cm³
Number = 12.965/0.01235 = 1050 spheres

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Surface Areas & Volumes 📦