📦 Cuboid & Cube
CUBOID (l × b × h)
TSA = 2(lb + bh + hl)
LSA = 2(l + b) × h
Volume = l × b × h
Diagonal = √(l² + b² + h²)
CUBE (side = a)
TSA = 6a²
LSA = 4a²
Volume = a³
Diagonal = a√3
🎯 Board Tip: "Open box" means no lid → TSA minus one face. "Room walls" means only lateral surface → use LSA.
🧮 Cuboid Calculator
Q: A cuboidal box has dimensions 60 cm × 40 cm × 50 cm. Find its TSA and volume.
TSA = 2(60×40 + 40×50 + 50×60)
= 2(2400 + 2000 + 3000)
= 2 × 7400 = 14800 cm²
Volume = 60 × 40 × 50 = 120000 cm³ = 120 litres
🛢️ Right Circular Cylinder
CYLINDER (radius r, height h)
CSA = 2πrh
Unrolled → rectangle: 2πr × h
TSA = 2πr(h + r)
Volume = πr²h
HOLLOW CYLINDER (outer R, inner r)
Volume = πh(R² − r²)
🧮 Cylinder Calculator
Q: A cylindrical pillar has radius 20 cm and height 3.5 m. Find the cost of painting its CSA at ₹12/m².
r = 20 cm = 0.2 m, h = 3.5 m
CSA = 2 × 22/7 × 0.2 × 3.5 = 2 × 22/7 × 0.7 = 4.4 m²
Cost = 4.4 × 12 = ₹52.80
🍦 Right Circular Cone
CONE (radius r, height h, slant height l)
l = √(r² + h²)
CSA = πrl
TSA = πr(l + r)
Volume = ⅓ πr²h
Volume = ⅓ × cylinder volume with same r and h
FRUSTUM (radii r₁, r₂, height h)
l = √(h² + (r₂−r₁)²)
CSA = π(r₁+r₂)l
TSA = π(r₁+r₂)l + πr₁² + πr₂²
Vol = ⅓πh(r₁²+r₂²+r₁r₂)
🧮 Cone Calculator
💡 Why ⅓? It takes exactly 3 cones to fill a cylinder of the same radius and height. This can be demonstrated experimentally by pouring water!
Q: A conical tent has radius 7 m and height 24 m. Find the cost of canvas at ₹100/m² (ignore stitching margins).
l = √(7² + 24²) = √(49 + 576) = √625 = 25 m
CSA = 22/7 × 7 × 25 = 550 m²
Cost = 550 × 100 = ₹55,000
🔮 Sphere & Hemisphere
SPHERE (radius r)
Surface Area = 4πr²
= 4 × (area of great circle). No CSA/LSA distinction.
Volume = ⁴⁄₃ πr³
HEMISPHERE (radius r)
CSA = 2πr² (dome only)
TSA = 3πr² (dome + flat base)
Volume = ⅔ πr³
🧮 Sphere / Hemisphere Calculator
Q: A dome of a building is hemispherical with radius 2.8 m. Find the cost of painting it at ₹5/m².
CSA = 2πr² = 2 × 22/7 × 2.8 × 2.8 = 2 × 22/7 × 7.84 = 49.28 m²
Cost = 49.28 × 5 = ₹246.40
🧩 Combination of Solids
Key Principle: When two solids are joined, the contact surface is hidden. TSA = sum of visible surfaces (subtract joined faces). Volume = sum of individual volumes.
🚀 Cone on Cylinder
TSA = CSA(cyl) + CSA(cone) + area(base)
Volume = πr²h₁ + ⅓πr²h₂
💊 Capsule (Cyl + 2 Hemi)
TSA = 2πrh + 4πr² (no flat ends)
Volume = πr²h + ⁴⁄₃πr³
🍦 Cone on Hemisphere
TSA = πrl + 2πr² (flat faces cancel)
Volume = ⅓πr²h + ⅔πr³
Q: A wooden article is made by scooping out a hemisphere of radius 3.5 cm from each end of a cylinder of radius 3.5 cm and height 10 cm. Find TSA.
TSA = CSA(cylinder) + 2 × CSA(hemisphere)
= 2πrh + 2 × 2πr²
= 2 × 22/7 × 3.5 × 10 + 4 × 22/7 × 3.5²
= 220 + 154
= 374 cm²
♻ Conversion of Solids
Golden Rule: When a solid is melted and recast into another shape, volume is conserved. Surface area changes!
1
Write volume formula for the original solid
2
Write volume formula for the new solid
3
Set V₁ = V₂ (or V₁ = n × V₂ for multiple objects)
4
Solve for the unknown dimension or count
🧮 Melting & Recasting Calculator
Sphere melted into small spheres
Q: A metallic sphere of radius 4.2 cm is melted and recast into a cylinder of radius 6 cm. Find the height of the cylinder.
V(sphere) = V(cylinder)
⁴⁄₃ πr³ = πR²h
⁴⁄₃ × (4.2)³ = (6)² × h
⁴⁄₃ × 74.088 = 36 × h
98.784 = 36h
h = 98.784 / 36 = 2.744 cm
Q: Water flows at 10 m/min through a pipe of diameter 1.4 cm into a conical vessel of diameter 40 cm and depth 24 cm. How long will it take to fill the vessel?
Pipe: r = 0.7 cm, speed = 1000 cm/min
Volume/min = πr² × speed = 22/7 × 0.49 × 1000 = 1540 cm³/min
Cone: R = 20 cm, H = 24 cm
Volume = ⅓ × 22/7 × 400 × 24 = 10057.14 cm³
Time = 10057.14 / 1540 ≈ 6.53 minutes ≈ 6 min 32 sec