4. (Why cannot a convex polyedral angle be formed by combining regular hexagons, heptagons, or octagons ?) Hence only five regular convex polyedrons are possible, tetraedrons, octaedrons, and icosaedrons, having equilateral triangles as faces; hexaedrons having squares as faces; and dodecaedrons, having regular pentagons as faces. 616. To construct the regular polyedrons, 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 strips of paper along them. (Compare diagrams on page 270.) CYLINDERS 617. DEF. A cylindrical surface is a curved surface generated by a moving straight line which continually intersects a given MA fixed curve and is always parallel to a given straight line not in the same plane with the curve. 618. 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. 619. 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 II lines included between || planes. 620. DEF. A circular cylinder is a cylinder whose bases are circles. 621. DEF. A right cylinder is a cylinder whose elements are perpendicular to the bases. 622. DEF. An oblique cylinder is one whose elements are oblique to the bases 623. DEF. The altitude of a cylinder is the perpendicular distance between the bases. 624. DEF. A cylinder of revolution is a right circular cylinder because it may be generated by a rectangle revolving about one of its sides as an axis. 625. DEF. Similar cylinders of revolution are cylinders generated by similar rectangles revolving about homologous sides as axes. 626. DEF. A tangent line to a cylinder is a straight line, which touches the lateral surface in one point but does not intersect it. A tangent plane to a cylinder is a plane which contains one element of the cylinder and but one, and does not intersect the cylinder. 627. 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. 628. DEF. A prism is circumscribed about a cylinder when its lateral edges are parallel to the elements of the cylinder and its bases are circumscribed about the bases of the cylinder. 629. DEF. A section of a cylinder is the figure formed when the cylinder is intersected by a plane; a right section is a section formed by a plane perpendicular to the elements. PROPOSITION XXIV. THEOREM 630. Every section of a cylinder made by a plane passing through an element is a parallelogram. Hyp. ABCD is a section of cylinder AC, made by plane through element AB. To prove ABCD is a parallelograin. Proof. Any straight line through D in plane AC II to AB is an element of the cylindrical surface. (Why?) Since this line is in the plane AC and is an element of the cylindrical surface, it must be their intersection, and therefore coincides with DC. .. DC is a straight line Il to AB. Also AD is a straight line Il to BC. (Why?) :. ABCD is a parallelogram. Q.E.D. 631. Cor. Every section of a right cylinder made by a plane passing through an element is a rectangle. Ex. 1127. The altitude of a right cylinder is 12 inches and the radius of the base is 6 inches. Find the area of a section made by a plane passing through an element and perpendicular to a radius at a point whose distance from the center measured on this radius is (a) two inches. (6) five inches. Ex. 1128. What is the area of the figure formed in the preceding exercise when the plane passes through the center of the base ? |