Geom12point1 97

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Geom12point1 97

  1. 1. Chapter 12 - Surface Area & Volume of Solids <ul><li>Objectives: </li></ul><ul><li>Identify types of solids </li></ul><ul><li>Calculate surface area & volume of: </li></ul><ul><li>Prisms </li></ul><ul><li>Pyramids </li></ul><ul><li>Cylinders </li></ul><ul><li>Cones </li></ul><ul><li>Spheres </li></ul>
  2. 2. 12.1 Exploring Solids Objectives: Use properties of Polyhedra Use Euler’s Theorem
  3. 3. Polyhedra <ul><li>A polyhedron is a solid that is bounded by polygons, called faces , that enclose a single region of space. </li></ul>
  4. 4. Polyhedra <ul><li>An edge of a polyhedron is a line segment formed by the intersection of two faces. </li></ul><ul><li>A vertex of a polyhedron is a point where 3 or more edges meet. </li></ul><ul><li>The plural of polyhedron is polyhedra or polyhedrons . </li></ul>
  5. 5. Are these polyhedra?
  6. 6. Types of Solids <ul><li>Prism </li></ul><ul><li>Pyramid </li></ul><ul><li>Cone </li></ul><ul><li>Cylinder </li></ul><ul><li>Sphere </li></ul>
  7. 7. More Terms <ul><li>A polyhedron is regular if all its faces are congruent regular polygons. </li></ul><ul><li>A polygon is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. </li></ul><ul><li>If this segment goes outside the polyhedron, then the polyhedron is nonconvex , or concave . </li></ul>
  8. 8. Cross Sections <ul><li>Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section . </li></ul><ul><li>Did anyone see the Museum of Science exhibition Body Worlds ? </li></ul><ul><li>Describing cross sections </li></ul>
  9. 9. Platonic Solids <ul><li>There are 5 regular polyhedra, called Platonic solids after Greek mathematician and philosopher Plato. </li></ul><ul><ul><li>Tetrahedron 4 faces </li></ul></ul><ul><ul><li>Cube 6 faces </li></ul></ul><ul><ul><li>Octahedron 8 faces </li></ul></ul><ul><ul><li>Dodecahedron 12 faces </li></ul></ul><ul><ul><li>Icosahedron 20 faces </li></ul></ul>
  10. 10. How many vertices & edges? <ul><li>Tetrahedron 4 faces </li></ul><ul><li>6 vertices, 6 edges </li></ul><ul><li>Cube 6 faces </li></ul><ul><li>8 vertices, 12 edges </li></ul><ul><li>Octahedron 8 faces </li></ul><ul><li>6 vertices, 12 edges </li></ul><ul><li>Dodecahedron 12 faces </li></ul><ul><li>20 vertices, 30 edges </li></ul><ul><li>Icosahedron 20 faces </li></ul><ul><li>12 vertices, 30 edges </li></ul>
  11. 11. Euler’s Theorem <ul><li>The number of faces (F), vertices (V) and edges (E) of a polyhedron are related by the formula F + V = E + 2 </li></ul>
  12. 12. Example 6, p. 722 <ul><li>A soccer ball resembles a polyhedron with 32 faces, 20 regular hexagons and 12 regular pentagons. </li></ul><ul><li>How many vertices? </li></ul><ul><li>Each of the 20 hexagons has 6 sides and each of the 12 pentagons has 5 sides. </li></ul><ul><li>Each edge of the soccer ball is shared by 2 polygons. </li></ul><ul><li>So, the total number of edges is: E = 1/2 (6*20 + 5*12) = 90 </li></ul>
  13. 13. Example 6, p. 722 <ul><li>E = 1/2 (6*20 + 5*12) = 90 </li></ul><ul><li>F + V = E + 2 </li></ul><ul><li>32 + V = 90 + 2 </li></ul><ul><li>V = 60 </li></ul>

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