ICSE Class 10 Maths: Mensuration — Surface Area & Volume Notes 2026

Cylinder — Formulae

• Curved surface area (CSA) = 2πrh; Total surface area (TSA) = 2πr(h+r)
• Volume = πr²h
• Hollow cylinder: volume = π(R²−r²)h; TSA includes two annular rings

Cone — Formulae

• Slant height l = √(h²+r²)
• CSA = πrl; TSA = πr(l+r)
• Volume = ⅓πr²h; cone holds one-third of volume of cylinder with same r and h

Sphere and Hemisphere

• Sphere: SA = 4πr²; volume = (4/3)πr³
• Hemisphere: CSA = 2πr²; TSA = 3πr²; volume = (2/3)πr³
• Solid sphere cut into hemisphere: flat circular face exposed with area πr²

Frustum of a Cone

• Frustum: portion between two parallel cuts through a cone
• CSA = π(R+r)l where l = √[h² + (R−r)²]; TSA = π(R+r)l + π(R²+r²)
• Volume = (πh/3)(R² + r² + Rr)

Conversion Between Shapes

• Melting and recasting: volume remains constant (density same material)
• Sphere melted to form n small spheres: (4/3)πR³ = n × (4/3)πr³
• Cylinder melted to cone: equate volumes to find new dimensions

ICSE Problems — Combined Solids

• Cone on hemisphere: total volume and surface area; bucket (frustum) capacity
• Toy made of hemisphere and cone: find paint required (surface area)
• Water in cylindrical tank: draining into conical vessel — height of water in cone

ICSE Exam Tips — Mensuration

• Always check which surface area is asked: CSA or TSA
• Use π = 22/7 unless calculator allowed; show substitution clearly
• Draw diagram for combined solids; label all given dimensions