ICSE Class 10 Maths: Mensuration 3D — Sphere, Cone, Cylinder Notes

Cylinder Formulas

• Curved surface area (CSA) = 2πrh; Total surface area (TSA) = 2πr(r+h)
• Volume = πr²h; r = radius of base, h = height
• Hollow cylinder: volume = π(R²−r²)h; TSA includes inner curved surface, outer curved surface, and two ring-shaped ends

Cone Formulas

• Slant height l = √(r²+h²); CSA = πrl; TSA = πr(r+l)
• Volume = (1/3)πr²h; one-third of cylinder with same base and height
• Frustum (truncated cone): formed when cone is cut parallel to base; CSA = π(r₁+r₂)l; Volume = (πh/3)(r₁²+r₁r₂+r₂²)

Sphere Formulas

• Surface area = 4πr²; Volume = (4/3)πr³
• Hemisphere: curved surface area = 2πr²; TSA = 3πr² (includes flat base circle πr²)
• Hollow sphere/shell: volume = (4/3)π(R³−r³) where R = outer radius, r = inner radius

Combined Solid Shapes

• Toy: cone on hemisphere; total volume = V_cone + V_hemisphere; total surface = CSA cone + CSA hemisphere
• Bullet: cylinder + cone; total SA = CSA cylinder + CSA cone + one circle base
• ICSE frequently: calculate cost of painting, material needed, weight if density given

Conversions and Practical Applications

• Melted and recast: volume of original = volume of new shape; equate and solve for unknown dimension
• Rain water: volume of water collected = area of catchment × depth of rainfall (unit conversion needed)
• Cost problems: cost = surface area × rate (for painting/polishing) or volume × density × cost/kg (for material)

Units and π Value

• Use π = 22/7 unless question specifies 3.14 or other value; consistent throughout solution
• Units: if r in cm and h in cm, volume in cm³; 1 litre = 1000 cm³; 1 m³ = 1000 litres
• Express final answer with appropriate units; ICSE may deduct for missing units

ICSE Exam Strategy

• Mensuration: 6–8 marks in ICSE maths paper; combined shapes most common; always draw diagram
• Show each formula, substitute values, calculate; multi-step problems need clear organisation
• Common mistake: using diameter where radius required; always check r vs d in problem statement