PROBLEMS OF COMPUTATION 1243 Compute the lateral area of a cone of revolution of which the radius of the base is 3 inches and the altitude is 4 inches. 1244 Compute the total area of a right circular cone of which the radius of the base is 7 feet and the slant height is 23 feet. 1245 Find the volume of a cone of revolution, the radius of its base being 48 inches, and its slant height being 73 inches. 1246 Find the volume of a circular cone whose altitude is 10 feet, and the diameter of whose base is 10 feet. 1247 The radii of the bases of a frustum of a cone of revolution are 6 inches and 4 inches, and the slant height is 5 inches. Find the lateral area. 1248 The diameters of the bases of a frustum of a circular cone are 18 feet and 12 feet, and the altitude is 16 feet. Find the volume. 1249 The radii of the bases of a frustum of a right circular cone are 7 feet and 4 feet, and its altitude is 5 feet. Find the altitude of the cone from which the frustum is cut off. 1250 The altitudes of two similar cones of revolution are as 2:1. What is the ratio of their total areas? of their volumes ? 1251 The volumes of two similar cones are 216 cu. ft. and 512 cu. ft. What is the height of the first, if the height of the second is 12 ft. ? 1252 The altitude of a cone is 36 inches. How far from the vertex must it be cut by a plane so that the frustum shall be equivalent to half the cone ? 1253 How many square inches of tin are required to make a funnel, if the slant height is 9 inches, and the diameters of its bases are 6 inches and 1 inch, allowing inch for seam ? 1254 The volumes of two similar cones of revolution are as 27:125. Compare their convex surfaces. 1255 Of two cylinders of revolution, the altitude of one is three times the altitude of the other. Compare their convex surfaces and their volumes. 1256 Find the edge of a cube equivalent to a right circular cylinder whose diameter is 6 ft. and whose altitude is 10 ft. BOOK VIII THE SPHERE DEFINITIONS 745 A sphere is a solid bounded by a surface all points of which are equidistant from a point within called the center of the sphere. A sphere may be generated by the revolution of a semicircle about its diameter as an axis. 746 The radius of a sphere is a straight line drawn from the center to any point of the surface. The diameter of a sphere is a straight line drawn through the center and terminated by the surface. 747 A line or a plane is tangent to a sphere when it has one point only in common with the surface of the sphere. Two spheres are tangent to each other when their surfaces have one point only in common. 748 COROLLARY. All radii of a sphere are equal. All diameters of a sphere are equal. Spheres having equal radii are equal, and conversely. A point is within a sphere, on its surface, or without a sphere, when its distance from the center is less than, equal to, or greater than, the radius. PROPOSITION I. THEOREM 749 Every section of a sphere made by a plane is a circle. HYPOTHESIS. O is the center of a sphere, and ABC is any section made by a plane. CONCLUSION. The section ABC is a circle. PROOF Draw ODL to the section, and join O to any two points in the perimeter of the section, as A and B. Then OA =OB... DA = DB. § 748 That is, all points in the perimeter of the section ABC are equidistant from the point D... the section ABC is a circle. Q. E. D. 750 DEFINITION. A circle of a sphere is any section of the sphere made by a plane. 751 COROLLARY 1. The line joining the center of a sphere and the center of a circle of the sphere is perpendicular to the plane of the circle. 752 COROLLARY 2. Circles of a sphere equidistant from the center of the sphere are equal, and conversely. 753 COROLLARY 3. Of two circles unequally distant from the center of a sphere, the more remote is the smaller, and conversely. DEFINITIONS 754 A great circle of a sphere is a circle passing through the center of the sphere. A small circle of a sphere is a circle of the sphere not passing through the center of the sphere. 755 The axis of a circle of a sphere is the diameter of the sphere perpendicular to the circle. The poles of a circle are the extremities of its axis. 756 The distance between two points on the surface of a sphere is the length of the minor are of the great circle joining them. 757 COROLLARY 1. The center of a great circle coincides with the center of the sphere. 758 COROLLARY 2. All great circles of a sphere are equal. For their radii are radii of the sphere. § 757 759 COROLLARY 3. Any two great circles of a sphere bisect each other. For since they have the same center, they have the same diameter. 760 COROLLARY 4. A great circle bisects the sphere. For the two parts may be made to coincide. § 745 761 COROLLARY 5. An arc of a circle may be drawn through any three points on the surface of a sphere. For the three points determine a plane (§ 515) which cuts the sphere in a circle. § 749 762 COROLLARY 6. An arc of a great circle may be drawn through any two points on the surface of a sphere. For the two points on the surface and the center of the sphere determine a great circle. § 515 The extremities of a diameter and the center of the sphere, being in a straight line, do not determine a circle. 763 COROLLARY 7. If the planes of two great circles are perpendicular, each circle passes through the poles of the other. PROPOSITION II. THEOREM 764 All points in the circumference of a circle of a sphere are equidistant from either of its poles. HYPOTHESIS. P and P' are the poles of the circle ABC, and A, B, and C are any three points in the circumference. CONCLUSION. The arcs PA, PB, and PC are equal. Also, the arcs P'A, P'B, and P'C are equal. PROOF The straight lines PA, PB, and PC are equal. § 530 § 243 In like manner it may be proved that the arcs P'A, P'B, and P'C are equal. Q. E. D. 765 DEFINITIONS. The polar distance of a circle of a sphere is the distance measured on the surface of the sphere from a point in the circumference of the circle to its nearer pole. A quadrant in Spherical Geometry is one fourth the circumference of a great circle. 766 COROLLARY. The polar distance of a great circle is a quadrant. For let EDF and PDP' be great circles. Then O is their common center, and the right angle POD is measured by the quadrant PD. |