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Exploring Polyhedra: Hexahedra, Parallelepipeds, and Euler’s Formula

This lesson focuses on polyhedra, specifically classifying hexahedra and defining related terms. Students will learn about parallelepipeds, including theorems regarding their properties, such as the parallelism and congruence of opposite edges, and the bisecting nature of diagonals. Additionally, Euler’s formula, which relates the number of vertices, edges, and faces in convex polyhedra, will be introduced. Exercises will reinforce understanding through practical applications, ensuring mastery of these essential geometric concepts.

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Exploring Polyhedra: Hexahedra, Parallelepipeds, and Euler’s Formula

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  1. Lesson 10.5 Polyhedra pp. 434-438

  2. Objectives: 1. To classify hexahedra and define related terms. 2. To prove theorems for parallelpipeds. 3. To state and apply Euler’s formula.

  3. Definition A polyhedron is a closed surface made up of polygonal regions.

  4. Definition A parallelepiped is a hexahedron in which all faces are parallelograms. A diagonal of a hexahedron is any segment joining vertices that do not lie on the same face.

  5. AD is a diagonal parallelepiped A D B C

  6. AC is not a diagonal parallelepiped A D B C

  7. A D B C AB is an edge of the cube; AC is a diagonal of the square face of the cube; AD is a diagonal of the cube.

  8. Definition Opposite faces of a hexahedron are faces with no common vertices. Opposite edges of a hexahedron are two edges of opposite faces that are joined by a diagonal of the parallelepiped.

  9. parallelepiped F E A D H G B C ABCD & EFGH are opposite faces

  10. parallelepiped F E A D H G B C ABCD & CDFG are not opposite faces

  11. parallelepiped F E A D H G B C

  12. BC & EF are opposite edges parallelepiped F E A D H G B C

  13. BC & AD are not opposite edges parallelepiped F E A D H G B C

  14. Theorem 10.16 Opposite edges of a parallelepiped are parallel and congruent.

  15. Theorem 10.17 Diagonals of a parallelepiped bisect each other.

  16. Theorem 10.18 Diagonals of a right rectangular prism are congruent.

  17. Euler’s Formula V - E + F = 2 where V, E, and F represent the number of vertices, edges, and faces of a convex polyhedron respectively.

  18. Euler’s formula applies not only to parallelepipeds but to all convex polyhedra.

  19. Tetrahedron V = 4 E = 6 F = 4 V - E + F = 2 V = E = F = V - E + F =

  20. Octahedron V = E = F = V - E + F = V = 6 E = 12 F = 8 V - E + F = 2

  21. Homework pp. 437-438

  22. ►A. Exercises For each decahedron below, determine the number of faces, edges, and vertices. Check Euler’s formula for each. 7.

  23. 7.

  24. ►B. Exercises Each exercise below refers to a prism having the given number of faces, vertices, edges, or sides of the base. Determine the missing numbers to complete the table below. Draw the prism when necessary; find some general relationships between these parts of the prism to complete exercise 18.

  25. ►B. Exercises F V S E Example 14 24 12 36 13. 7 10 15. 7 17. 8

  26. 13. Faces (F) = 7 Vertices (V) = 10 Sides of the base (S) = Edges (E) = 5 15

  27. ►B. Exercises F V n E Example 14 24 12 36 13. 7 10 515 15. 7 17. 8 18. n

  28. 17. Faces (F) = 8 Vertices (V) = Sides of the base (S) = Edges (E) = 12 6 18

  29. ►B. Exercises F V n E Example 14 24 12 36 13. 7 10 515 15. 7 17. 8 12 6 18 18. n

  30. A E D B C ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 24. Find the area.

  31. D E C A B ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 25. Prove that A  B.

  32. ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 26. Find the distance between two numbers a and b on a number line.

  33. ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 27. True/False: Water contains helium or hydrogen.

  34. ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 28. When are the remote interior angles of a triangle complementary?

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