1 / 11

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

channer
Download Presentation

12.1 Exploring Solids

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related