Lesson 10.5 Polyhedra pp. 434-438

Lesson 10.5Polyhedra

pp. 434-438

Lesson 10.5Polyhedra

pp. 434-438

Objectives:1. To classify hexahedra and define

related terms.2. To prove theorems for

parallelpipeds.3. To state and apply Euler’s formula.

Objectives:1. To classify hexahedra and define

related terms.2. To prove theorems for

parallelpipeds.3. To state and apply Euler’s formula.

A A polyhedronpolyhedron is a closed is a closed surface made up of polygonal surface made up of polygonal regions.regions.

DefinitionDefinitionDefinitionDefinition

A A parallelepipedparallelepiped is a is a hexahedron in which all faces hexahedron in which all faces are parallelograms.are parallelograms.

A A diagonal of a hexahedrondiagonal of a hexahedron is is any segment joining vertices any segment joining vertices that do not lie on the same that do not lie on the same face.face.

DefinitionDefinitionDefinitionDefinition

parallelepipedparallelepiped

AA

BB CC

DD

AD is a diagonalAD is a diagonal

parallelepipedparallelepiped

AA

BB CC

DD

AC is not a diagonalAC is not a diagonal

AA

BB CC

DD

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.

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.

Opposite faces of a Opposite faces of a hexahedronhexahedron are faces with no are faces with no common vertices.common vertices.

Opposite edges of a Opposite edges of a hexahedronhexahedron are two edges of are two edges of opposite faces that are joined opposite faces that are joined by a diagonal of the by a diagonal of the parallelepiped.parallelepiped.

DefinitionDefinitionDefinitionDefinition

HH

parallelepipedparallelepiped

AA

BB CC

DD

ABCD & EFGH are opposite facesABCD & EFGH are opposite faces

EE FF

GG

HH

parallelepipedparallelepiped

AA

BB CC

DD

ABCD & CDFG are not opposite facesABCD & CDFG are not opposite faces

EE FF

GG

HH

parallelepipedparallelepiped

AA

BB CC

DD

EE FF

GG

HH

parallelepipedparallelepiped

AA

BB CC

DD

EE FF

GG

BC & EF are opposite edgesBC & EF are opposite edges

HH

parallelepipedparallelepiped

AA

BB CC

DD

EE FF

GG

BC & AD are not opposite edgesBC & AD are not opposite edges

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

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

Theorem 10.17Diagonals of a parallelepiped bisect each other.

Theorem 10.17Diagonals of a parallelepiped bisect each other.

Theorem 10.18Diagonals of a right rectangular prism are congruent.

Theorem 10.18Diagonals of a right rectangular prism are congruent.

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

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

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

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

V =

E =

F =

V - E + F =

V =

E =

F =

V - E + F =

V = 4

E = 6

F = 4

V - E + F = 2

V = 4

E = 6

F = 4

V - E + F = 2

TetrahedronTetrahedron

OctahedronOctahedron

V =

E =

F =

V - E + F =

V =

E =

F =

V - E + F =

V = 6

E = 12

F = 8

V - E + F = 2

V = 6

E = 12

F = 8

V - E + F = 2

Homeworkpp. 437-438Homeworkpp. 437-438

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

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

7.7.

►B. ExercisesEach 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.

►B. ExercisesEach 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.

F V S E

Example 14 24 12 36

13. 7 10

15. 7

17. 8

F V S E

Example 14 24 12 36

13. 7 10

15. 7

17. 8

►B. Exercises►B. Exercises

13.

Faces (F) = 7

Vertices (V) = 10

Sides of the base (S) =

Edges (E) =

13.

Faces (F) = 7

Vertices (V) = 10

Sides of the base (S) =

Edges (E) =

55

1515

F V n E

Example 14 24 12 36

13. 7 10 5 15

15. 7

17. 8

18. n

F V n E

Example 14 24 12 36

13. 7 10 5 15

15. 7

17. 8

18. n

►B. Exercises►B. Exercises

17.

Faces (F) = 8

Vertices (V) =

Sides of the base (S) =

Edges (E) =

17.

Faces (F) = 8

Vertices (V) =

Sides of the base (S) =

Edges (E) =

66

1818

1212

F V n E

Example 14 24 12 36

13. 7 10 5 15

15. 7

17. 8 12 6 18

18. n

F V n E

Example 14 24 12 36

13. 7 10 5 15

15. 7

17. 8 12 6 18

18. n

►B. Exercises►B. Exercises

■ Cumulative ReviewDo not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them.24. Find the area.

■ Cumulative ReviewDo not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them.24. Find the area.

AA

BB CC

DD

EE

■ Cumulative ReviewDo not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them.25. Prove that A B.

■ Cumulative ReviewDo not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them.25. Prove that A B.

AA BB

CC

DD EE

■ Cumulative ReviewDo 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.

■ Cumulative ReviewDo 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.

■ Cumulative ReviewDo 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.

■ Cumulative ReviewDo 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.

■ Cumulative ReviewDo 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?

■ Cumulative ReviewDo 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?