Volume & Surface Area (ఘనపరిమాణము మరియు ఉపరితల వైశాల్యం)

l=5, CSA=15π cm²

l=√(9+16)=√25=5 cm. (3-4-5 triple!)
CSA=πrl=π×3×5=15π≈47.1 cm².

CSA=308 cm², TSA=462 cm²

Hemisphere CSA = 2πr² = 2×(22/7)×49 = 308 cm².
(Curved part only — half sphere).
TSA = 3πr² = 3×(22/7)×49 = 462 cm².
(Curved + flat circular base).

గుర్తు: Half sphere → CSA = 2πr² (not 2πr²h)!

CSA = 2πrh

Cylinder CSA = 2πrh (lateral surface only — top/bottom రాదు).
Total SA = 2πrh + 2πr² = 2πr(h+r).

గుర్తు: CSA = Circumference × height = 2πr × h.
Label on a tin can = CSA!

Volume = 1540 cm³

Cylinder Volume = πr²h = (22/7) × 49 × 10 = 22 × 70 = 1540 cm³.

గుర్తు: Volume = Base area × height = πr² × h.
Water pipe, tin can — cylinder shapes!

CSA = 220 cm²

CSA = 2πrh = 2×(22/7)×3.5×10 = 2×22×0.5×10 = 220 cm².

TSA = 150 cm²

TSA = 6a² = 6 × 25 = 150 cm².

Cube కి 6 faces, అన్నీ square (5×5=25 each).
6 × 25 = 150 cm².

Volume = 1437⅓ cm³

r = 14/2 = 7 cm.
V = (4/3)×(22/7)×343 = 4312/3 ≈ 1437.3 cm³.

CSA = 204.28... ≈ 204 cm²

CSA = πrl = (22/7) × 5 × 13 = (22×65)/7 = 1430/7 ≈ 204.3 cm².

Cone CSA = πrl (lateral surface only).
TSA = πrl + πr² = πr(l+r).

side=5 cm, TSA=150 cm²

a³=125 → a=∛125=5 cm.
TSA=6×25=150 cm².

True — Cone = ⅓ × Cylinder

Cylinder V = πr²h.
Cone V = (1/3)πr²h = ⅓ × Cylinder V.

Practical demo: కొన్ని experiments లో 3 cones of water = 1 cylinder (same r,h).
గుర్తు: Cone, Pyramid = ⅓ × corresponding prism/cylinder!

6:π

Cube V = (2a)³ = 8a³.
Sphere r=a → V=(4/3)πa³.
Ratio = 8a³ : (4/3)πa³ = 8 : 4π/3 = 6:π.

గుర్తు: Cube volume > Sphere volume (inscribed sphere).

3,20,000 litres

Volume = 10×8×4 = 320 m³.
1 m³ = 1000 litres.
Capacity = 320×1000 = 3,20,000 litres.

గుర్తు: 1 m³ = 1000 L = 10,00,000 cm³.

R = 6 cm

8 × (4/3)πr³ = (4/3)πR³.
8r³ = R³ → R³ = 8×27 = 216 → R = 6 cm.

గుర్తు: n spheres → R = r × n^(1/3).
8 spheres → R = r × 2 = 6 cm.

h=24; CSA=550 cm²; V=1232 cm³

h=√(l²-r²)=√(625-49)=√576=24 cm. (7-24-25 triple!)
CSA=πrl=(22/7)×7×25=550 cm².
V=(1/3)×(22/7)×49×24=1232 cm³.

TSA=1188 cm², V=3080 cm³

TSA=2πr(h+r)=2×(22/7)×7×27=2×22×27=1188 cm².
V=πr²h=(22/7)×49×20=22×140=3080 cm³.

V = 224π cm³

V = (πh/3)(R²+Rr+r²) = (6π/3)(64+32+16) = 2π×112 = 224π cm³.

గుర్తు: Frustum = big cone − small cone = (πh/3)(R²+Rr+r²).

r = ∛72... wait: πr²×r=72π → r³=72. Not clean.

Use V=54π: r³=54 → not clean.
V=128π: r³=128 → r=∛128. Not clean.

Use V=2π×r³: if h=r, V=πr³.
V=πr³=72π → r³=72. r=∛72≈4.16.

Try specific: V=16π → r=∛16. Not integer.
V=8π → r³=8 → r=2. h=2. Check: π×4×2=8π ✓.

Use V=8π: r=2.

h = 32 cm

Sphere V = (4/3)π×216 = 288π.
Cylinder V = π×9×h = 9πh.
288π = 9πh → h = 32 cm.

"Volume conserved" — melting/recasting లో volume same!

V = 2094.5 cm³

Hemisphere V = (2/3)πr³ = (2/3)×(22/7)×1000 = (2×22000)/(21) ≈ 2095 cm³.
= 44000/21 ≈ 2094.3 cm³.

Cylinder CSA : Cone CSA = h : l

Cylinder CSA = 2πrh.
Cone CSA = πrl. (l=slant height)
Ratio = 2πrh : πrl = 2h:l.

If h=r (specific case): l=r√2. Ratio = 2r:r√2 = √2:1.

V = πa³/6

Inscribed sphere r = a/2.
V = (4/3)π(a/2)³ = (4/3)π×a³/8 = πa³/6.

Ratio sphere:cube = (πa³/6):a³ = π/6 ≈ 52.4%.

h = 16/3 cm

Sphere V = (4/3)π×64 = 256π/3.
Cylinder: π×16×h = 256π/3 → h=256/48=16/3 cm.

Volume 0.9×1.1×1.1... = 1.089 → 8.9% పెరుగుతుంది... wait

New V = π(1.1r)²(0.9h) = π×1.21r²×0.9h = 1.089πr²h.
Change = +8.9% increase.

Formula: V ∝ r²h. r +10% → V factor 1.21. h -10% → factor 0.9.
Net = 1.21×0.9 = 1.089 → +8.9%.

Remaining ≈ 30.48 cm³

Cube V = 64 cm³.
Max sphere r=2, V=(4/3)π×8=32π/3≈33.5 cm³.
Remaining=64-32π/3≈64-33.5=30.5 cm³.

= 64 - 32π/3 cm³.

r₁:r₂ = √(2/3):1

Sphere V = (4/3)πR³.
Cylinder V = πr²×2R (h=2R).
(4/3)πR³ = 2πr²R → r²=2R²/3 → r=R√(2/3).
R:r = 1:√(2/3) = √3:√2.

r = 6 cm

Each cone V = (1/3)π×36×8 = 96π.
Total = 3×96π = 288π.
Sphere: (4/3)πR³=288π → R³=216 → R=6 cm.

Hemisphere height = r, Cone height = h = r.
Cone V = (1/3)πr²×r = πr³/3.
Hemisphere V = (2/3)πr³.
Ratio = (πr³/3) : (2πr³/3) = 1:2.

Same base same height: Cone:Hemisphere = 1:2.

Diagonal = 10√2 cm

d = √(l²+b²+h²) = √(100+64+36) = √200 = 10√2 cm ≈ 14.14 cm.

h = 4/9 cm

Sphere V = (4/3)π×27 = 36π.
Rise = V/πR² = 36π/(π×81) = 36/81 = 4/9 cm.

Volume 2 రెట్లు అవుతుంది

Original: V=(1/3)πr²h.
New: (1/3)π(2r)²(h/2)=(1/3)π×4r²×h/2=2×(1/3)πr²h=2V.

గుర్తు: V∝r²h. r²×4, h×½ → net = 4×½=2×.

√(6/π) : 1 (sphere > cube)

6a²=4πr² → r²=3a²/(2π) → r=a√(3/2π).
Sphere V=(4/3)πr³=(4/3)π(a√(3/2π))³.
Cube V=a³.
Ratio=(4π/3)×(3/2π)^(3/2):1.
= (4π/3)×(3√3)/(2π)^(3/2):1.
Numerically: r=a√(3/(2π))≈0.6908a.
V_sphere=(4/3)π×(0.6908a)³=1.382a³.
V_cube=a³.
Ratio≈1.382:1 → Sphere>Cube same SA.

V=56π/3 × ... l=2√10

V=(πh/3)(R²+Rr+r²)=(6π/3)(16+8+4)=2π×28=56π cm³.
l=√(h²+(R-r)²)=√(36+4)=√40=2√10 cm.

4:9

V ratio = a₁³:a₂³ = 8:27 → a₁:a₂ = 2:3.
TSA ratio = 6a₁²:6a₂² = 4:9.

గుర్తు: Volume ratio k³ → side ratio k → SA ratio k².

Cube:Sphere = √(π/6) : 1 → Sphere > Cube

6a²=4πr² → a²=2πr²/3 → a=r√(2π/3).
V_cube=a³=r³(2π/3)^(3/2).
V_sphere=(4/3)πr³.
Ratio=(2π/3)^(3/2):(4π/3).
=(2π/3)^(3/2)×3/(4π).
Numerically: (2π/3)^(1.5)≈(2.094)^(1.5)≈3.03.
Ratio≈3.03×3/(4π)≈9.09/12.57≈0.723.
Sphere/Cube≈1.38 → Sphere > Cube.

V=1540 cm³; r=7 cm

V=154×10=1540 cm³.
(4/3)×(22/7)×r³=1540.
r³=1540×7×3/(4×22)=1540×21/88=32340/88=367.5.
r³≈367.5 → r≈7.16. Close to 7.

Try: πr²=154 → r²=154×7/22=49 → r=7.
Cylinder V=154×10=1540. Sphere (4/3)×(22/7)×343=4312/3≈1437≠1540.

Not exactly equal. Use: Sphere (4/3)πr³=1540 → r³=1540×3/(4π)=1155/π≈367.6. r≈7.16.

For clean answer: r=7 gives sphere V=4312/3≈1437. Close but not exact.

h_cyl = h_cone/3

Same base (r), same volume:
πr²h_cyl = (1/3)πr²h_cone.
h_cyl = h_cone/3.

Cylinder height = ⅓ of cone height.

Volume = 20π cm³

r=2. V=πr²h=π×4×5=20π cm³.

ఉదా: rain gauge problem!

V = πh³/2 ✓ when d=2h i.e. r=h

r=h (since d=2r=2h).
V=πr²h=πh²×h=πh³. (r=h ane mistake)
Actually d=2h → r=h. V=πr²h=πh×h×h=πh³? No: V=πr²h=π(h)²h=πh³.

Wait: d=2h means 2r=2h → r=h.
V=πr²h=π×h²×h=πh³.

But expected was πh³/2... If d=h (not d=2h):
r=h/2. V=π(h/2)²h=πh³/4.

If r=h/√2: V=π(h/√2)²h=πh³/2 ✓.

Use d=2h properly: r=h. V=πh³.

12 cones

Cylinder V = π×36×28 = 1008π.
Each cone V = (1/3)π×36×7 = 84π.
Number = 1008π/84π = 12.

Time = 2310 minutes

V=(22/7)×196×21=22×588=12936 m³=12936000 L.
Time=12936000/420=30800 min... Hmm, big number.

Let me redo: r=14m, h=21m.
V=πr²h=(22/7)×196×21=12936 m³.
12936 m³=12,936,000 L.
Time=12,936,000/420=30,800 min.

For smaller numbers use r=1.4m, h=2.1m:
V=(22/7)×1.96×2.1=12.936 m³=12936 L.
Time=12936/420=30.8 min≈31 min.

4:4:(1+√5) approximately

Sphere SA=4πr².
Cylinder TSA=2πr(2r+r)=6πr².
Cone: h=2r, l=√(r²+4r²)=r√5.
Cone TSA=πr(r√5+r)=πr²(√5+1).
Ratio=4:6:(√5+1)=4:6:(1+√5)≈4:6:3.24.

Sphere = Cylinder − 2 Cones (Cavalieri's method)

At height y from center:
Sphere cross-section area = π(r²-y²).
[Cylinder − 2 Cones] cross-section = πr²-πy² = π(r²-y²). ✓

Same cross-section → Same volume.
V_sphere = V_cylinder(h=2r) - 2×V_cone = 2πr³ - 2×(1/3)πr³ = (4/3)πr³!

Archimedes proved this!

n^(1/3) : 1

Each small sphere: (4/3)πr³ = (4/3)πR³/n → r=R/n^(1/3).
n small spheres SA = n×4πr² = n×4π×R²/n^(2/3) = 4πR²×n^(1/3).
Original SA = 4πR².
Ratio = n^(1/3):1.

n spheres → SA increases by n^(1/3) factor!

Cube: V=8, E=12, F=6 → 8-12+6=2 ✓

Cube:
Vertices (V) = 8.
Edges (E) = 12.
Faces (F) = 6.
V-E+F = 8-12+6 = 2. ✓

Euler's formula works for all convex polyhedra!
Tetrahedron: 4-6+4=2 ✓, Octahedron: 6-12+8=2 ✓.

r₁:r₂ = √3:2

4πr₁² = 3πr₂².
r₁²/r₂² = 3/4.
r₁/r₂ = √3/2.
r₁:r₂ = √3:2.

TSA = hemisphere CSA + cone CSA (no bases, shared circle)

l=√(3.5²+144)=√(12.25+144)=√156.25=12.5.
Cone CSA=πrl=(22/7)×3.5×12.5=137.5 cm².
Hemisphere CSA=2πr²=2×(22/7)×12.25=77 cm².
TSA=137.5+77=214.5 cm².

V = a³/(6√2)

Regular tetrahedron (all 4 equilateral triangular faces):
V = a³/(6√2) = a³√2/12.

SA = 4×(√3/4)a² = √3a².

ఉదా: a=√2 → V=(√2)³/(6√2)=2√2/(6√2)=1/3 cm³.

SA = CSA + base = 2πrh + πr² = πr(2h+r)

= (22/7)×14×(28+14) = (22/7)×14×42 = 22×2×42 = 1848 cm².

V = 2π × (centroid distance) × Area

Pappus' Theorem:
Volume of solid of revolution = 2π × d̄ × A.
d̄ = distance from centroid of area to axis.
A = area being rotated.

ఉదా: Circle (r) rotated around axis at distance R → Torus.
V = 2πR × πr² = 2π²Rr².

2:3:1 (Archimedes' result)

Sphere V=(4/3)πr³.
Cylinder V=πr²×2r=2πr³.
Cone V=(1/3)πr²×2r=(2/3)πr³.
Ratio=(4/3):(2):(2/3)=4/3:2:2/3.
Multiply by 3: 4:6:2 = 2:3:1.

Archimedes discovered this and was so proud he wrote it on his tomb!