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12.1 Exploring Solids

12.1 Exploring Solids. Geometry. 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

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12.1 Exploring Solids

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  1. 12.1 Exploring Solids Geometry

  2. 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

  3. 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

  4. 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.

  5. More definitions • Regular polyhedron – all faces are , regular polygons. i.e. a cube • Convex polyhedron – all the polyhedra we’ve 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

  6. 5 regular polyhedra • Also called platonic solids. • Turn to page 796 for good pictures at the top of the page. • Tetrahedron – 4 equilateral Δ faces • Cube (hexahedron) – 6 square faces • Octahedron – 8 equilateral Δ faces • Dodecahedron – 12 pentagon faces • Icosahedron – 20 equilateral Δ faces

  7. Thm: Euler’s Theorem The # of faces (F), vertices (V), & edges (E) are related by the equation: F + V = E + 2 Remember the first example? Let’s flashback…

  8. 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

  9. Ex: A solid has 10 faces: 4 Δs, 1 square, 4 hexagons, & 1 octagon. How many edges & vertices does the solid have? 4 Δs = 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)

  10. Ex: A geodesic dome (like the silver ball at Epcot Center) is composed of 180 Δ faces. How many edges & vertices are on the dome? 180 Δs = 180(3) = 540 edges 540  2 = 270 edges F + V = E + 2 180 + V = 270 + 2 180 + V = 272 V = 92 vertices

  11. Assignment

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