Class 9 Mathematics Chapter 11 Surface Areas and Volumes — Formulas & Key Points

All Surface Area Formulas — Quick Reference Table

• Cuboid: LSA = 2h(l + b); TSA = 2(lb + bh + hl) where l = length, b = breadth, h = height
• Cube (all edges equal 'a'): LSA = 4a²; TSA = 6a²
• Right Circular Cylinder: CSA = 2πrh; TSA = 2πr(r + h) where r = radius of base, h = height
• Right Circular Cone: CSA = πrl; TSA = πr(l + r) where l = slant height = √(r² + h²)
• Sphere: Surface Area = 4πr² (no distinction between curved and total since sphere has one continuous surface)
• Hemisphere: CSA = 2πr²; TSA = 3πr² (includes the flat circular base)

All Volume Formulas — Quick Reference Table

• Cuboid: Volume = l × b × h (product of length, breadth, height)
• Cube: Volume = a³ where a is the edge length
• Right Circular Cylinder: Volume = πr²h
• Right Circular Cone: Volume = (1/3)πr²h (exactly one-third of cylinder volume)
• Sphere: Volume = (4/3)πr³
• Hemisphere: Volume = (2/3)πr³ (exactly half of sphere volume)

Key Terms and Definitions

• Cuboid: A rectangular box with six rectangular faces; opposite faces are equal and parallel.
• Cube: A special cuboid where all edges are equal; all six faces are congruent squares.
• Right Circular Cylinder: A solid with two parallel circular bases and a curved surface perpendicular to the bases.
• Right Circular Cone: A solid with one circular base and a curved surface tapering to a single apex (vertex) directly above the centre of the base.
• Sphere: A perfectly round solid where every point on the surface is equidistant from the centre; radius r is that distance.
• Hemisphere: Exactly half of a sphere, created by slicing through the centre; has one curved surface and one flat circular base.
• Slant Height (l): For a cone, l = √(r² + h²); the straight-line distance from apex to any point on the circumference of the base.
• Lateral / Curved Surface Area: The area of the curved part only, excluding top and bottom circular or rectangular faces.
• Total Surface Area: The sum of all outer surfaces including bases and curved parts.
• Volume: The amount of three-dimensional space inside a solid, measured in cubic units.

Important Constants and Values

• π ≈ 22/7 (use when radius or diameter is 7, 14, 21, etc.)
• π ≈ 3.14 or 3.1416 (use for other values or calculator-based problems)
• 1 m = 100 cm; 1 m² = 10,000 cm²; 1 m³ = 1,000,000 cm³
• 1 litre = 1000 cm³ = 0.001 m³
• Density = Mass / Volume (may appear in application problems)

Memory Tricks and Mnemonics

• Cuboid TSA mnemonic: 'Two Love Birds Have Lovely Hearts' → 2(lb + bh + hl)
• Cube: Everything is 'square'—edge², face area a², six faces so 6a², volume a³.
• Cylinder: '2 pi r' appears in both CSA (2πrh) and TSA (2πr(r + h)).
• Cone: 'One-third' for volume; slant height l is the hypotenuse √(r² + h²).
• Sphere: 'Four' for surface (4πr²), 'four-thirds' for volume ((4/3)πr³).
• Hemisphere: Half the sphere volume; CSA is half sphere surface; TSA adds base so factor becomes 3.

Common Sign, Unit and Notation Mistakes

• Always write units: cm², m² for area; cm³, m³ for volume. No units = mark deduction.
• Convert all measurements to the same unit before calculating (e.g. convert metres to centimetres).
• Radius vs. diameter: if diameter is given, radius r = d/2. Substituting diameter into r causes four-fold error in area, eight-fold in volume.
• For cone volume, do NOT forget the factor 1/3. It is (1/3)πr²h, not πr²h.
• For cone surface area, use slant height l in CSA = πrl, but perpendicular height h in volume.
• Sphere surface area is 4πr², not 2πr². Hemisphere TSA is 3πr², not 2πr².
• Square the radius in area and volume formulas: πr² and πr²h, not πr and πrh.
• Do not cancel π or numerical factors prematurely; keep them until the final step to avoid rounding errors.

Solved Mini-Example 1 — Cuboid Total Surface Area

Solved Mini-Example 2 — Cylinder Volume and Capacity

Solved Mini-Example 3 — Cone Slant Height and Surface Area

One-Glance Last-Minute Revision Box

• Cuboid: TSA=2(lb+bh+hl), Volume=lbh
• Cube: TSA=6a², Volume=a³
• Cylinder: CSA=2πrh, TSA=2πr(r+h), Volume=πr²h
• Cone: CSA=πrl, TSA=πr(l+r), Volume=(1/3)πr²h, l=√(r²+h²)
• Sphere: SA=4πr², Volume=(4/3)πr³
• Hemisphere: CSA=2πr², TSA=3πr², Volume=(2/3)πr³
• Units: area in cm² or m²; volume in cm³ or m³; 1 m³=1,000,000 cm³; 1 litre=1000 cm³
• Key trick: Cone & pyramid volumes are (1/3) × base area × height
• Always write units; convert all to same unit; radius = diameter/2

How CBSETUTOR.ai Helps You Master Surface Areas and Volumes

• Photo-upload solving: snap your NCERT exercise question and get a detailed worked solution instantly.
• Video explanations for every Surface Areas and Volumes formula, showing real exam-style applications.
• Unlimited practice questions auto-generated at varying difficulty levels to build speed and accuracy.
• One flat price ₹999/month for Classes 6-12, all subjects including Mathematics, Science and Social Science.
• 3-day free trial with no credit card required—test the platform risk-free before committing.