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. 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. 9, 10, 17. Ex. 1583. The three sides of the base of a pyramid are respectively 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 the base 30°. Find the volume. 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 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. 647. DEF. A section of a cylinder is the figure formed if the cylinder is intersected by a plane; a right section is a section formed by a plane perpendicular to the elements. PROPOSITION XXIII. THEOREM 648. Every section of a cylinder made by a plane passing through an element is a parallelogram. Given ABCD, a section of cylinder AC, made by plane through element AB. To prove ABCD is a parallelogram. Proof. Through D, in plane AC, draw a line || AB. 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. Also .. DC is a straight line || AB. AD is a straight line || BC. .. ABCD is a parallelogram. (Why?) Q. E. D. 649. COR. Every section of a right cylinder made by a plane passing through an element is a rectangle. PROPOSITION XXIV. THEOREM 650. The bases of a cylinder are congruent. Given ABC and A'B'C', the bases of the cylinder BC. Proof. Let A, B, and C be any three points in the lower base, and AA', BB', CC' the corresponding elements. Place the upper base upon the lower base so that A'B' coincides with AB. Then 'must coincide with C. But as c'is any point in the perimeter of the upper base, all points in the upper base coincide with the corresponding points in the lower base. .. the bases are congruent. Q. E. D. 651. COR. 1. Any two parallel sections cutting all the elements of a cylinder are congruent. 652. COR. 2. Any section of a cylinder parallel to the base is congruent to the base. 653. DEF. The area of the lateral surface of a cylinder is the limit which the lateral area of an inscribed prism ap |