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# Geom12point1 97

## by herbison on Jun 09, 2008

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## Geom12point1 97Presentation Transcript

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