Plane and solid geometry . s. A polyhedron of four o y e ron faces is called a tetrahedron; one of six faces, a hexahedron;one of eight faces, an octahedron; one of twelve faces, a do-decahedron ; one of twenty faces, an icosahedron; etc. Ex. 1250. How many diagonals has a tetrahedron ? a hexahedron ? Ex. 1251. What is the least number of faces that a polyhedron canhave ? edges ? vertices ? Ex. 1252. How many edges has a tetrahedron ? a hexahedron ? anoctahedron ? Ex. 1253. How many vertices has a tetrahedron ? a hexahedron ?an octahedron ? Ex. 1254. If ^ represents the number of edges, F the
Plane and solid geometry . s. A polyhedron of four o y e ron faces is called a tetrahedron; one of six faces, a hexahedron;one of eight faces, an octahedron; one of twelve faces, a do-decahedron ; one of twenty faces, an icosahedron; etc. Ex. 1250. How many diagonals has a tetrahedron ? a hexahedron ? Ex. 1251. What is the least number of faces that a polyhedron canhave ? edges ? vertices ? Ex. 1252. How many edges has a tetrahedron ? a hexahedron ? anoctahedron ? Ex. 1253. How many vertices has a tetrahedron ? a hexahedron ?an octahedron ? Ex. 1254. If ^ represents the number of edges, F the number offaces, and V the number of vertices in each of the polyhedrons mentionedin Exs. 1252 and 1253, show that in each case E-\-2 = V-\- F. Thisresult is known as Eulers theorem. 348 344 SOLID GEOMETRY Ex. 1255. Show that in a tetrahedron S=(^V~2]where S is the sum of the face angles and Fis the numUer^f ver Ex. 1256. Does the formula, S = {Y — 2) 4 right angles,^Kold fora hexahedron ? an octahedron ? a dodecahedron ?. 720. Def. A regular polyhedron is a polyhedron all ofwhose faces are equal regular polygons, and all of whose polyhe-dral angles are equal. 721. Questions. How manTOQiiilateral triangles can meet to forma polyhedral angle (§ 712) ? Then •>/ .i\ is the greatest number of regularpolyhedrons possible having equilateral triangles as faces ? What is thegreatest number of regular polyhedrons .possible having squares as faces ?having regular pentagons as faces ? Can a regular polyhedron have asfaces regular polygons of more than five sides ? why ? What, then, isthe maximum number of kinds of regular polyhedrons possible ? 722. From the questions in § 721, the^ student has doubtless^drawn the conclusion that not more than five kinds of regularpolyhedrons exist. He should convince himself that thesefive are possible by actually making them from cardboard asindicated below:
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
