Lesson 10.5 Polyhedra pp. 434-438
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Transcript of Lesson 10.5 Polyhedra pp. 434-438
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Lesson 10.5Polyhedra
pp. 434-438
Lesson 10.5Polyhedra
pp. 434-438
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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.
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A A polyhedronpolyhedron is a closed is a closed surface made up of polygonal surface made up of polygonal regions.regions.
DefinitionDefinitionDefinitionDefinition
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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
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parallelepipedparallelepiped
AA
BB CC
DD
AD is a diagonalAD is a diagonal
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parallelepipedparallelepiped
AA
BB CC
DD
AC is not a diagonalAC is not a diagonal
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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.
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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
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HH
parallelepipedparallelepiped
AA
BB CC
DD
ABCD & EFGH are opposite facesABCD & EFGH are opposite faces
EE FF
GG
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HH
parallelepipedparallelepiped
AA
BB CC
DD
ABCD & CDFG are not opposite facesABCD & CDFG are not opposite faces
EE FF
GG
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HH
parallelepipedparallelepiped
AA
BB CC
DD
EE FF
GG
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HH
parallelepipedparallelepiped
AA
BB CC
DD
EE FF
GG
BC & EF are opposite edgesBC & EF are opposite edges
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HH
parallelepipedparallelepiped
AA
BB CC
DD
EE FF
GG
BC & AD are not opposite edgesBC & AD are not opposite edges
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Theorem 10.16Opposite edges of a parallelepiped are parallel and congruent.
Theorem 10.16Opposite edges of a parallelepiped are parallel and congruent.
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Theorem 10.17Diagonals of a parallelepiped bisect each other.
Theorem 10.17Diagonals of a parallelepiped bisect each other.
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Theorem 10.18Diagonals of a right rectangular prism are congruent.
Theorem 10.18Diagonals of a right rectangular prism are congruent.
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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.
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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.
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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
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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
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Homeworkpp. 437-438Homeworkpp. 437-438
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►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.
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7.7.
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►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.
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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
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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
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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
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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
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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
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■ 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
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■ 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
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■ 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.
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■ 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.
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■ 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?