PROBLEM X. To find the volume of a spherical sector. Multiply the area of the zone which serves for its base by one third of the radius of the sphere.* Ex. 1. Required the volume of a spherical sector, the altitude of the zone which serves for a base being 12, and the diameter of the sphere being 30. Ans. 12 x 30 × 3·1416 × 5. Ex. 2. Required the volume of a spherical sector, a great section of the zone base being an arc of 40°, and the diameter of the sphere being 100. PROBLEM XI. To find the volume of a spherical segment. RULE I. From three times the diameter of the sphere take double the height of the segment; then multiply the remainder by the square of the height, and the product by the decimal 5236 for the content. (See Schol. to Prop. XIV., Sol. Geom.) RULE II. To three times the square of the radius of the segment's base add the square of its height; then multiply the sum by the height, and the product by 5236, for the content. RULE III. When the segment has two bases, multiply the half sum of the parallel bases by the altitude, and add the volume of the sphere of which this altitude is the diameter. *The spherical sector may be supposed to be made up of an infinite number of indefinitely small cones, each having an evanescent portion of the surface of the zone base for a base, and the radius of the sphere for an altitude. The sum of these will be measured by the sum of their bases, or the zone multiplied by one third their com mon altitude, or the radius of the sphere. When the zone becomes the whole surface of the sphere the sector becomes the whole solid sphere. Note that one third the radius is one sixth the diameter. = For, Rule 2, observe that d2 surface of sphere (Prob. 7), and 5236. The rule is similar for a spherical pyramid having a spherical polygon for a base and the center of the sphere for a vertex. K Ex. 1. To find the content of a spherical segment of two feet in height cut from a sphere of 8 feet in diameter. Ans. 41.888. Ex. 2. What is the solidity of the segment of a sphere, its height being 9, and the diameter of its base 20? Ans. 1795 4244. EXERCISES IN MENSURATION.* 1. Transform a given parallelogram into another of double the altitude which shall have a given angle. 2. To transform a triangle into another of the same base and given vertical angle. 3. To construct a triangle of given base, vertical angle, and area. 4. The same, except the altitude instead of the base given. 5. To construct a triangle similar to a given triangle, and equal to a given square. 6. A triangle with given angles at the base, and equal to a given rectangle. 7. The same, when the base, vertical angle, and rectangle of the other two sides are given. 8. The same, when the base, the altitude, and the product of the two sides. 9. The same, when the altitude, the area, and the ratio of one of the sides to the base. 10. The same, when the ratio of the base and altitude, the vertical angle and the area. 11. Make a regular hexagon equivalent to a given polygon. 12. To construct a figure similar to a given figure, and its area having to that of the given figure a given ratio. 13. A quadrilateral capable of being inscribed, in which two adjacent angles, the angle which its diagonals make with each other and its area, are given. 14. A quadrilateral that may be inscribed, in which three angles and the arca are given. 15. A circle equal to the sum of several circles. 16. A square in a given semicircle. 17. A circle equal to the ring between two circles. 18. A quadrant equal to a given semicircle. * Many of these will conveniently admit the application of logarithms. 19. A sextant equal to a given quadrant. 20. To determine the side of an equilateral triangle, the area of which is 73-45. 21. Also, of a regular hexagon, the area of which is 168. 22. The side of a regular pentagon is 21-7. What is that of another half as large? 23. To find the radius of a semicircle equal to a triangle whose base is 14, and altitude 9. 24. What is the diameter of a circle equal to a trapezoid, of which the base is 17.4, the opposite side 12·7, and the altitude 10.08! 25. To find the content of a regular octagon when the radius of its inscribed circle is equal to 12. 26. Of a regular decagon when the radius of the inscribed circle is equal to 17.2. 27. How large is the angle at the center of a circular sector, the area of which is equal to that of an equilateral triangle whose side is 14, the radius of the sector being 8? 28. To determine the side of a square which shall be equal to a sector whose are is 18, and radius 7·5. 29. To determine the diameter of a circle which shall be equal to the segment of a circle whose radius is 120, and arc 135°. 30. To determine the radii of the inscribed and circumscribed circles of a triangle whose sides are 10, 12, and 14. 31. To find the convex surface of a regular hexagonal prism, the longest diameter of which is 2r, and height h. 32. Of a regular hexagonal pyramid with the same data. 33. Of a zone of one base, the radius of which is r, and altitude h. 34. Of a spherical sector, the chord of which =c, and rad. sphere =r. 35. To find the volume of a solid generated by the revolution of a sector of a circle about a line through the center, and exterior to the sector. 36. Find the volume of the solid generated by the revolution of any triangle about one of its sides. 37. Find the volume of a solid generated by the revolution of the segment of a circle about a line passing through the center of the circle, and exterior to the segment. 38. Prove that the surfaces of two spheres are as the squares, and the volumes as the cubes of their radii. 39. Find the volume left of a cylinder after a spherical segment having one base equal to that of the cylinder, and the same altitude with the cylinder, has been abstracted. 40. The altitude and surface of a regular hexagonal prism, of which the greatest diameter is 18, and the volume of which is equal to that of a regular triangular pyramid, of which the base side is equal to 8, and altitude 20. 41. In a quadrangular and hexagonal prism each side of the bases is 7, the height 13. What is the ratio of their volumes and surfaces? 42. To find the altitude of a regular quadrangular pyramid, the side of whose base is 28.7, and volume equal to that of a rectangular parallelopipedon whose edges are 13, 17, and 23. 43. The ratio of two homologous edges of two similar polyhedrons is 5:7. To find the ratio of their surfaces and volumes. 44. The ratio of the volumes of two similar polyhedrons is 14:29. To find that of their homologous edges. 45. What is the ratio of the surfaces of two regular pyramids, the one triangular, the other quadrangular, if the base in both is 24, and the altitude 7? 46. A regular tetrahedron, the edge of which is 15, has the third of its altitude cut off by a plane parallel to the base; required the volume of the frustum left. Also, the surface. 47. About a sphere of 16 inches radius a polyhedron is circumscribed, containing 20,800 cubic inches. What is the area of the surface of the latter? 48. A cylinder and cone have their radii 14 and 8, their altitudes 6 and 9. What is the ratio of their volumes and surfaces? 49. Find the radius of a sphere equal to a cube, the diagonal of which is 17.22. 50. Also, of a sphere equal to a regular octahedron, the diagonal of which is 31.5. 51. Find the radius of an inscribed sphere in a regular tetrahedron, the edge of which is a. 52. The same for an octahedron. 53. Find the ratio of the surfaces of a regular tetrahedron and inscribed sphere. 54. The same for an octahedron and sphere. 55. What is the ratio of a hemisphere to a cone of the same base and altitude? 56. Find the ratio of the solids generated by a triangle and rectangle revolving about a common base, and the altitude of the former being double that of the latter. 57. To find the volume of a spherical segment when the radius is 5.86, and the arc of a great section 162° 14'. 58. Find the base of a square pyramid which shall contain a cubic yard, and the altitude of which shall be 1 foot. 59. The sides of the base of a tetrahedron are 12, 15, 17, its altitude 9. Required its volume. Ans. 263 248. 60. A regular tetrahedron contains 19.683 cubic yards. Required its edges and surface. Ans. Edge 5.50705, surface 52-5289. 61. Required the volume of a frustum of a regular triangular pyramid, the larger base of which has 0-9 for its side, and the smaller base 0.4, and of which the lateral edge is 0·5. Ans. 0.078371. 62. Given the volume of a sphere equal to 1843-086278 to find its radius. Ans. 7.61. 63. Given the edge of a cube 0:36. Required the volume of the circumscribed sphere. Ans. 0.126937. 64. Find the area of a spherical triangle, the angles of which are respectively 85 grades, 17', 103, 35', 678, 49', the radius of the sphere being 1.54. Ans. 2.0865. 65. There is a crucible in the form of a conic frustum, the bottom of which is 0.03 in diameter, the top 0.06, and the altitude 0.08; this crucible contains a quantity of melted metal, the surface of which is 0:05 in diameter: it is required to make a sphere of it. diameter of the proper mold? What is the Ans. 0.507444. 66. Given the side or apophthegm of a cone 25·15, and its height 17.3, to find its convex surface and volume. 67. Find the quantity of glass in a lens, of which the diameters of the surfaces are 0·03, and the thickness of the lens 0·004.* Ans. 0.000001422094. 68. Supposing the earth to be perfectly spherical, and a quarter of the meridian to be expressed by 10,000,000; find the expression for its radius, the area of its surface, its volume and weight, supposing the mean density of the earth to be 5.6604.† *This solid is a double segment of a sphere. This number is the result of the experiments of Sir Francis Bailey, given in the xivth vol. of the Memoirs of the Royal Ast. Soc. of Lond., 1844. |