Regular Polyhedra: Formulas

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also see Defining Geometric Figures

 Regular Polyhedron     A solid, three-dimensional figure each face of which is a regular polygon with     equal sides and equal angles. Every face has the same number of vertices, and the     same number of faces meet at every vertex. An inscribed (inside) sphere touches     the center of every face, and a circumscribed sphere (outside) touches every vertex.     There are five and only five of these figures, also called the Platonic Solids:     the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Number of vertices: v Number of edges: e Number of faces: f Edge: a Radius of circumscribed sphere: R Radius of inscribed sphere: r Surface area: S Volume: V Dihedral angle between faces: delta (in degrees) To read about regular polyhedra, visit The Geometry Center: Regular Polyhedra Mathematics Encyclopedia: The Platonic Solids Eric Swab: Cube as the Base To explore polyhedra via an MJA applet, try Suzanne Alejandre's Studying Polyhedra For many more facts and links, see Alexander Bogomolny's Regular polyhedra For a Java applet showing many other polyhedra, see John N. Huffman's Crystallographic Polyhedra

 Tetrahedron (see also Irregular Tetrahedron) A three-dimensional figure with 4 equilateral triangle faces, 4 vertices, and 6 edges. v = 4, e = 6, f = 4 a = (2 sqrt[6]/3)R r = (1/3)R R = (sqrt[6]/4)a S = sqrt(3)a2 V = (sqrt[2]/12)a3 delta = arccos(1/3) = 70o 32' h = height or altitude h = (sqrt[6]/3)a

 Cube A three-dimensional figure with 6 square faces, 8 vertices, and 12 edges. v = 8, e = 12, f = 6 a = (2 sqrt[3]/3)R r = (sqrt[3]/3)R R = (sqrt[3]/2)a r = (1/2)a S = 6 a2 V = a3 delta = arccos(0) = 90o

See Ask Dr. Math: Surface Area and Volume: Cubes and Prisms

 Octahedron A three-dimensional figure with 8 equilateral triangle faces, 6 vertices, and 12 edges. v = 6, e = 12, f = 8 a = sqrt(2)R r = (sqrt[3]/3)R R = (sqrt[2]/2)a r = (sqrt[6]/6)a S = 2 sqrt(3)a2 V = (sqrt[2]/3)a3 delta = arccos(-1/3) = 109o 28'

 Dodecahedron A three-dimensional figure with 12 regular pentagon faces, 20 vertices, and 30 edges. v = 20, e = 30, f = 12 a = ([sqrt(5)-1]sqrt[3]/3)R r = (sqrt[75+30 sqrt(5)]/15)R R = (sqrt[3][1+sqrt[5])/4)a r = (sqrt[250+110 sqrt(5)]/20)a S = 3 sqrt(25+10 sqrt[5])a2 V = ([15+7 sqrt(5)]/4)a3 delta = arccos(-sqrt[5]/5) = 116o 34'

 Icosahedron A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges. v = 12, e = 30, f = 20 a = (sqrt[50-10 sqrt(5)]/5)R r = (sqrt[75+30 sqrt(5)]/15)R R = (sqrt[10+2 sqrt(5)]/4)a r = (sqrt[42+18 sqrt(5)]/12)a S = 5 sqrt(3)a2 V = (5[3+sqrt(5)]/12)a3 delta = arccos(-sqrt[5]/3) = 138o 11'

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