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Section 8 4 Polyhedrons Spheres Section 8 4 Nack Jones 1 Polyhedron 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 2 Euler 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 2 Where V of vertices F of faces E of edges Example 1 p 420 Section 8 4 Nack Jones 3 Regular 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 4 Spheres Three Characteristics 1 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 5 Surface 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 r Theorem 8 4 3 The volume V of a sphere with radius of length r is given by V 4 3 r3 Example 4 6 p 424 Solids of Revolution Revolving a semi circle sphere Revolving circle around line torus p 425 426 Section 8 4 Nack Jones 6