lesson 10.5 polyhedra pp. 434-438
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Lesson 10.5 Polyhedra pp. 434-438. Objectives: 1.To classify hexahedra and define related terms. 2.To prove theorems forparallelpipeds. 3.To state and apply Euler’s formula. Definition. A polyhedron is a closed surface made up of polygonal regions. Definition. - PowerPoint PPT PresentationTRANSCRIPT
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?