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Section 8.4 Polyhedrons & Spheres

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Section 8.4 Polyhedrons & SpheresPolyhedronEuler’s EquationRegular PolyhedronSpheresSurface Area and Volume of a SphereSection 8.4 Nack/Jones 1Section 8.4Polyhedrons & SpheresSection 8.4 Nack/Jones 2Polyhedron•Plural: polyhedrons or polyhedra•A solid bounded by plane regions.•The faces of the polyhedrons are polygons•The edges are the line segments common to these polygons •Vertices are the endpoints of the edges•Convex: Each face determines a plane for which all remaining faces lie on the same side of the plane. p.420.•Concave: Two vertices and the line segment containing them lies in the exterior of the polyhedron.Section 8.4 Nack/Jones 3Euler’s Equation•Theorem 8.4.1: The number of vertices V, the number of edges, E, and the number of faces F of a polyhedron are related by the equation. V + F = E + 2Where V = # of vertices F = # of faces E = # of edgesExample 1 p. 420Section 8.4 Nack/Jones 4Regular Polyhedron•A regular polyhedron is a convex polyhedron whose faces are congruent regular polygons arranged in such a way that adjacent faces form congruent dihedral angles (the angle formed when two edges intersect).Section 8.4 Nack/Jones 5Spheres•Three Characteristics1. A sphere is the set of all points at a fixed distance r from a given point O. Point O is known as the center of the sphere.2. A sphere is the surface determined when a circle (or semicircle) is rotated about any of its diameters.3. A sphere is the surface that represents the theoretical limit of an “inscribed” regular polyhedron whose number of faces increase without limit.Section 8.4 Nack/Jones 6Surface Area and Volume of a Sphere•Theorem 8.4.2: The surface area S of a sphere whose radius has length r is given by S = 4r²•Theorem 8.4.3: The volume V of a sphere with radius of length r is given by V =4/3 r3Example 4 – 6 p. 424•Solids of Revolution:–Revolving a semi circle = sphere–Revolving circle around line = torus p. 425 -


Section 8.4 Polyhedrons & Spheres

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