12'1 Exploring Solids

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12.1 Exploring Solids

• p. 719

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Defns. for 3-dimensional figures

• Polyhedron a solid bounded by polygons that
  enclose a single region of shape. (no curved
  parts no openings!)
• Faces the polygons (or flat surfaces)
• Edges segments formed by the intersection of 2
  faces
• Vertex point where three or more edges intersect

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Ex Is the figure a polyhedron? If so, how many
faces, edges, vertices are there?
Yes, F V E
No, there are curved parts!
5 6 9
Yes, F V E
7 7 12
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Types of Solids

• Prism 2 ? faces (called bases) in ?? planes.
  i.e. first example
• Pyramid has 1 base, all other edges connect at
  the same vertex. i.e. last example
• Cone like a pyramid, but base is a circle.
• Cylinder 2 ?? circle bases.
• or
• Sphere like a ball.

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More definitions

• Regular polyhedron all faces are ?, regular
  polygons. i.e. a cube
• Convex polyhedron all the polyhedra weve seen
  so far are convex.
• Concave polyhedron
• caves in
• Cross section the intersection of a plane
  slicing through a solid. Good picture on p.720

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5 regular polyhedra

• Also called platonic solids.
• Turn to page 721 for good pictures at the top of
  the page.
• Tetrahedron 4 equilateral ? faces
• Cube (hexahedron) 6 square faces
• Octahedron 8 equilateral ? faces
• Dodecahedron 12 pentagon faces
• Icosahedron 20 equilateral ? faces

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Thm 12.1 Eulers Theorem

• The of faces (F), vertices (V), edges (E) are
  related by the equation
• F V E 2
• Remember the first example?
• Lets flashback

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Ex How many faces, edges, vertices are there?
F V E
5 6 9
F V E 2 5 6 9 2 11 11
F V E
7 7 12
F V E 2 7 7 12 2 14 14
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Ex A solid has 10 faces 4 ?s, 1 square, 4
hexagons, 1 octagon. How many edges vertices
does the solid have?

• 4 ?s 4(3) 12 edges
• 1 square 4 edges
• 4 hexagons 4(6) 24 edges
• 1 octagon 8 edges
• F V E 2
• 10 V 24 2
• 10 V 26
• V 16 vertices

48 edges total
But each edge is shared by 2 faces, so they have
each been counted twice!
This means there are actually 24 edges on the
solid. (? by 2)
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Ex A geodesic dome (like the silver ball at
Epcot Center) is composed of 180 ? faces. How
many edges vertices are on the dome?

• 180 ?s 180(3) 540 edges
• 540 ? 2 270 edges
• F V E 2
• 180 V 270 2
• 180 V 272
• V 92 vertices

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Assignment