633. DEF. Similar polyhedrons are polyhedrons that have the same number of faces similar each to each and similarly placed, and have their homologous polyhedral angles equal. PROPOSITION XXI. THEOREM 634. The volumes of two similar tetrahedrons are to each other as the cubes of their homologous edges. Given v and v', the volumes of the similar tetrahedrons S-ABC and T-EFG. Ex. 1577. In the diagram for Prop. XXII, if SA == 3 and TE=2, find the ratio of V to V'. Ex. 1578. In the same diagram, find TF, if SB = 2, and V ; V' = 1 : 2, REGULAR POLYHEDRONS 635. DEF. A regular polyhedron is a polyhedron all of whose faces are equal regular polygons, and all of whose polyhedral angles are equal. It is proved in Prop. XXII that only five regular polyhedrons are possible, viz. the tetrahedron, the octahedron, the cube, the icosahedron, and the dodecahedron. (For diagrams see pp. 340 and 374.) PROPOSITION XXII. PROBLEM 636. To determine the number of regular convex polyhedrons possible. A convex polyhedral angle must have at least three faces, and the sum of its face angles must be less than 360°. (523) 1. A convex polyhedral angle may be formed by combining three, four, or five equilateral triangles. Since each angle of an equilateral A is 60°, the sum of six such angles is 360°, a sum greater than that of the face angles of a convex polyhedral angle. Hence three regular convex polyhedrons are possible with equilateral A as faces. 2. A convex polyhedral angle may be formed by combining three squares, but not by using four or more squares. (Why?) Hence one regular convex polyhedron is possible with squares as faces. 3. A convex polyhedral angle may be formed by combining three regular pentagons, but not by using four, etc., pentagons. (Why?) Hence one regular convex polyhedron can be formed having regular pentagons as faces. 4. It is impossible to form a convex polyhedral angle by combining hexagon, heptagon, octagon, etc. (Why ?) Hence only five regular convex polyhedrons are possible; the tetrahedron, the octahedron, the icosahedron (having equi lateral triangles as faces), the hexahedron or the cube (having squares as faces), and the dodecahedron (having regular pentagons as faces). 637. To construct the regular polyhedrons, draw on stiff paper or cardboard the following diagrams. Cut partly through the paper along the dotted lines. Fold over and hold the edges in contact by pasting the narrow strips. A Ex. 1579. If three equal lines AA', BB', and CC' have their midpoints at a point O, and each line is perpendicular to the other two, then the points A, B, C, A', B', C', determine a regular octahedron. (The lines AA', BB', and CC' are called the axes of the octahedron.) Ex. 1580. If a side of an octahedron equals 2 in., find the length of the axes (AA', etc.). Ex. 1581. If a side of an octahedron equals 4 in., find the volume and the surface of the solid. Ex. 1582. The three sides of the base of a pyramid are respectively 10, 17, and 21. Find the volume if the altitude is 5. Ex. 1583. The three sides of the base of a pyramid are respectively 9, 10, 17. Find the volume if a lateral edge is 20, and its projection upon the base equals 12. Ex. 1584. A lateral edge of a pyramid equals 10, and its inclination to the base is 30°. Find the area of the base if the volume of the pyramid is 100. Ex. 1585. The base of a pyramid is a rhombus whose diagonals are respectively 10 and 12. Find the volume if the altitude is 6. Ex. 1586. The diagonals of a parallelopiped divide the figure into six equivalent pyramids. Ex. 1587. If any point within a parallelopiped be joined to the 8 vertices, 6 pyramids are formed, of which the sum of any opposite two is equal to the sum of any other opposite two. Ex. 1588. Each edge of a triangular pyramid is equal to 10. Find the volume. Ex. 1589. The perimeter of the triangular base of a regular pyramid is 30. Find the volume if the altitude is 12. Ex. 1590. The base of a pyramid is a parallelogram of base 10 and altitude 8. Find the volume if a lateral edge is equal to 6, and forms with the base an angle of 45°. Ex. 1591. The base of a pyramid is a rectangle having sides respectively equal to a and b. A lateral edge is equal to c, and is inclined to Find the volume. the base 30°. Ex. 1592. Find the altitude of a pyramid of base b, equivalent to another pyramid of base a and altitude h. CYLINDERS 638. DEF. A cylindrical surface is a surface generated by a moving straight line that continually intersects a fixed curve and is always parallel to a fixed straight line not in the same plane with the given curve. 639. DEF. The generatrix of the surface is the moving straight line; the directrix is the given curve; and an element of the surface is the moving line in any of its positions. 640. DEF. A cylinder is a solid bounded by a cylindrical surface and two parallel planes; the bases of a cylinder are the parallel planes; and the lateral surface is the cylindrical surface. The elements of a cylinder are equal since they are Il lines included between || planes. 641. DEF. A circular cylinder is a cylinder whose bases are circles. 642. DEF. A right cylinder is a cylinder whose elements are perpendicular to the bases. 643. DEF. An oblique cylinder is one whose elements are oblique to the bases. 644. DEF. The altitude of a cylinder is the perpendicular distance between the bases. 645. DEF. A straight line is tangent to a cylinder, if the line touches the lateral surface in one point but does not intersect it if produced. A plane is tangent to a cylinder if it contains one element of the cylinder and does not intersect the cylinder. 646. DEF. A prism is inscribed in a cylinder when its lateral edges are elements of the cylinder and its bases are inscribed in the bases of the cylinder. |