Chapter 11 Extending Geometry

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Chapter 11Extending Geometry

• Section 11.4
• Three-Dimensional Figures

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The Regular Polyhedra

• A polyhedron is a collection of polygons joined
  to enclose a region of space.
• Parts of a polyhedron
• Faces
• Vertices
• Edges

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Platonic Solids

• Regular polyhedra (Platonic solids) have edges of
  equal length and the arrangement of polygons at
  each vertex is the same.
• Called Platonic solids because Plato associated
  earth, air, fire, water, and creative energy with
  these solids and used them in his description of
  the universe.
• The ancient Greeks used prefixes that indicated
  the number of faces to name these polyhedra.
• Tetra- four
• Hexa- six
• Octa- eight
• Dodeca- twelve
• Icosa- twenty

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Five Platonic Solids

• Regular Tetrahedron
• Regular Hexahedron (Cube)
• Regular Octahedron
• Regular Dodecahedron
• Regular Icosahedron

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Five Platonic Solids
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Investigation
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Eulers Formula

• Eulers Formula F V E 2
• Named after Swiss mathematician, Leonhard Euler.

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Prisms

• A prism is a polyhedron with a pair of congruent
  faces called bases, that lie in parallel planes.
  
• The vertices of the bases are joined to form the
  lateral faces (which are always parallelograms)
  of a prism.
• Prisms are named according to the shapes of their
  bases.

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Parts of a Prism
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Prisms

• If the lateral edges of a prism are perpendicular
  to its bases, the prism is a right prism.
• If the lateral edges of a prism are not
  perpendicular to the bases, the prism is an
  oblique prism.

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Right and Oblique Prisms
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Pyramids

• A pyramid is a polyhedron formed by connecting
  the vertices of a polygon, called the base, to a
  point not in the plane of the polygon, called an
  apex.
• The lateral faces of a pyramid are always
  triangles.
• The segment from the vertex perpendicular to the
  base is called the altitude.
• Pyramids are named according to the shapes of
  their bases.

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Right Regular Pyramids

• Right regular pyramid base is a regular
  polygon, altitude is perpendicular to base at its
  center, and lateral faces are isosceles
  triangles. The height of the isosceles
  triangular lateral face is called the slant
  height.

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Parts of a Pyramid
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Examples of Pyramids
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Cylinders, Cones, and Spheres

• Cylinders, cones, and spheres are all solids with
  curved surfaces.
• In a cylinder and cone, the bases are circles.
• The line through the centers of the bases of a
  cylinder is called the axis of the cylinder.

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Cylinders, Cones, and Spheres
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Visualizing Polyhedra

• A pattern or planar net for a polyhedron is an
  arrangement of polygons that can be folded to
  form the polyhedron.

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What Polyhedra Can Be Made With These Patterns?