Chapter 13: Surface Areas and Volumes
Key Formulas:
• Sphere: SA=4πr², Volume=4/3 πr³
• Hemisphere: CSA=2πr², TSA=3πr², Volume=2/3 πr³
• Cylinder: CSA=2πrh, TSA=2πr(r+h), Volume=πr²h
• Cone: CSA=πrl, TSA=πr(l+r), Volume=1/3 πr²h
• Frustum: Volume=1/3 πh(r₁²+r₂²+r₁r₂)
• CSA of frustum=π(r₁+r₂)l where l=√(h²+(r₁-r₂)²)
Exercise 13.1
Q1. 2 cubes each of volume 64cm² are joined end to end. Find surface area of resulting cuboid.
SolutionSide of each cube = √√64 = 4cm
Cuboid: l=8cm, b=4cm, h=4cm
TSA = 2(8×4+4×4+8×4)
= 2(32+16+32)
= 2(80) = 160 cm²
Q2. A vessel is in form of hollow hemisphere mounted by a hollow cylinder. Diameter 14cm, total height 13cm. Find inner curved surface area.
SolutionRadius = 7cm
Height of cylinder = 13-7 = 6cm
CSA of cylinder = 2π×7×6 = 84π cm²
CSA of hemisphere = 2π×7² = 98π cm²
Total CSA = 84π+98π = 182π cm²
Q3. A toy is in form of cone on hemisphere. Radius 3.5cm, total height 15.5cm. Find total surface area.
SolutionCone height = 15.5-3.5 = 12cm
Slant height l = √(12²+3.5²) = √(144+12.25) = √156.25 = 12.5cm
CSA of cone = π×3.5×12.5 = 43.75π
CSA of hemisphere = 2π×3.5² = 24.5π
Total TSA = 43.75π+24.5π = 68.25π cm²
Q5. A cubical block of side 7cm is surmounted by hemisphere. Find surface area of solid.
SolutionTSA of cube = 6 × 7² = 294 cm²
CSA of hemisphere = 2π × (7/2)² = 2π × 49/4 = 24.5π
Base area of hemisphere (circle on top) = π × 49/4 = 12.25π
Total = 294 - 12.25π + 24.5π
= 294 + 12.25π
= 294 + 38.48 ≈ 332.48 cm²
Q6. A cubical block of side 7cm is surmounted by hemisphere. Find volume of solid.
SolutionVolume of cube = 7³ = 343 cm³
Volume of hemisphere = 2/3 π × (7/2)³
= 2/3 × π × 343/8
= 343π/12
= 90.01...
Total = 343 + 343π/12 ≈ 433.02 cm³