Grade 6 Lecture: Relationship of Volumes of Prisms, Pyramids, and Other 3D Figures

Lecture Overview

• Objective: Determine volume relationships between:
  • Rectangular prism and pyramid
  • Cylinder and cone
  • Cylinder and sphere
• Outcome: Understand how these relationships help in solving volume-related problems for 3D figures.

Key Concepts

Volume of Rectangular Prism

• Volume formula: Length × Width × Height
• Example:
  • One layer contains 24 cubic units
  • 8 layers in total
  • Total volume = 192 cubic units

Volume Relationship: Rectangular Prism and Pyramid

• A pyramid with the same base and height as a rectangular prism has 1/3 the volume of the prism.
• Example:
  • Prism volume = 192 cubic units
  • Pyramid volume = 192 / 3 = 64 cubic units
  • If pyramid volume = 64 cubic units, prism volume = 64 × 3 = 192 cubic units

Volume Relationship: Cylinder and Cone

• A cone with the same base and height as a cylinder has 1/3 the volume of the cylinder.
• Example:
  • Cylinder volume = 288 cubic units
  • Cone volume = 288 / 3 = 96 cubic units
  • If cone volume = 78 cubic units, cylinder volume = 78 × 3 = 234 cubic units

Volume Relationship: Cylinder and Sphere

• A sphere with the same base and height as a cylinder has 2/3 the volume of the cylinder.
• Example:
  • Cylinder volume = 42 cubic units
  • Sphere volume = 2/3 × 42 = 28 cubic units
  • If sphere volume = 51 cubic units, cylinder volume = 51 × 3/2 = 76.5 cubic units

Learning Tasks

Task 1: Volume Calculation

1. Cylinder & Cone
   • Cylinder volume = 78 cm³
   • Cone volume = 78 / 3 = 26 cm³
2. Prism & Pyramid
   • Pyramid volume = 36 cm³
   • Prism volume = 36 × 3 = 108 cm³
3. Cylinder & Sphere
   • Cylinder volume = 198 m³
   • Sphere volume = 198 × 2/3 = 132 m³

Task 2: Model Creation

• Create a rectangular prism and pyramid with the same dimensions using cardboard or a folder.
• Objective: Demonstrate it takes three pyramids to fill one prism.

Task 3: Multiple-Choice Questions

1. Volume of pyramid = 1/3 volume of prism
2. Volume of cone = 1/3 volume of cylinder
3. Number of cones to fill sphere = ?
4. Cone volume with cylinder volume = 120 m³
5. Cube edge length if volume = 27 in³

Conclusion

• Practice these volume relationships through hands-on activities and tasks.
• Share your findings and understanding with classmates for collaborative learning.