Measurement
Volume and surface area of complex solids
Volume and Surface Area of Complex Solids
Grade 11 extends measurement to spheres, cones and combinations of shapes.
• Sphere: V = (4/3)πr³, SA = 4πr²
• Cone: V = (1/3)πr²h, SA = πr² + πrl (l = slant height)
• Combined shapes: add or subtract volumes and surface areas appropriately.
Example
Sphere and Cone
Sphere with r = 6 cm:
V = (4/3)π(6)³ = 288π ≈ 904.8 cm³
SA = 4π(6)² = 144π ≈ 452.4 cm²
Cone with r = 4 cm, h = 9 cm, l = √(16+81) = √97 ≈ 9.85 cm:
V = (1/3)π(4)²(9) = 48π ≈ 150.8 cm³
Example
Effect of Scaling
If dimensions are multiplied by factor k:
• Length × k
• Area × k²
• Volume × k³
Double the radius of a sphere:
New volume = 2³ × original = 8 times the original volume.
Note
Remember
Slant height l = √(r² + h²) for a cone. The factor effect is crucial: linear → ×k, area → ×k², volume → ×k³. Always state units clearly.
Key Vocabulary
SphereA perfectly round 3D shape (like a ball)
ConeA 3D shape with a circular base tapering to a point
Slant heightThe distance along the sloping surface of a cone
Scale factorThe multiplier that changes all dimensions
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