1. Lesson 10.5
Polyhedra pp

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. parallelepiped A D B C
AD is a diagonal

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

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

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

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 = 6
E = 12
F = 8
V - E + F = 2
V =
E =
F =
V - E + F =

21. Homework pp

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
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
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
18. n

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

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

32. ■ Cumulative Review
Do not solve exercises 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 27. True/False: Water contains helium or hydrogen.
Do not solve exercises 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 below, but write (in complete sentences) what you would do to solve them.
28. When are the remote interior angles of a triangle complementary?