parallel to one side of the base and to the opposite lateral edge, intersects its faces in a parallelogram. 14. The lateral surface of a pyramid is greater than the base. Project the vertex on the base, etc. 15. The four middle points of two pairs of opposite edges of a tetraedron are in one plane, and at the vertices of a parallelogram. (617). 16. The three lines joining the middle points of the three pairs of opposite edges of a tetraedron intersect in a point which bisects them all. 17. In a tetraedron, the planes passed through the three lateral edges and the middle points of the edges of the base intersect in a straight line. The intersections of the planes with the base are medials of the base; .. etc. 18. The lines joining each vertex of a tetraedron with the point of intersection of the medial lines of the opposite face all meet in a point, which divides each line in the ratio 1: 4. NOTE.—This point is the centre of gravity of the tetraedron. 19. The plane which bisects a diedral angle of a tetraedron divides the opposite edge into segments which are proportional to the areas of the adjacent faces. See (303), (369). 20. The volume of a truncated triangular prism is equal to the product of the area of its lower base by the perpendicular upon the lower base let fall from the intersection of the medial lines of the upper base. 21. The volume of a truncated parallelopiped is equal to the product of the area of its lower base by the perpendicular from the centre of the upper base upon the lower base. 22. The volume of a truncated parallelopiped is equal to the product of a right section by one-fourth the sum of its four lateral edges. See (640). 23. Any plane passed through the centre of a parallelopiped divides it into two equivalent solids (640). a 24. The portion of a tetraedron cut off by a plane parallel to one of its faces is a tetraedron similar to the whole tetraedron. 25. When two tetraedrons have a diedral angle of the one equal to a diedral angle of the other, and the faces including these angles similar each to each, and similarly placed, the tetraedrons are similar. 26. Two polyedrons composed of the same number of tetraedrons, similar each to each, and similarly placed, are similar. NUMERICAL EXERCISES. 27. Find the lateral area of a right prism whose altitude is 14 inches and perimeter of the base 16 inches. Ans. 224 square inches. . 28. Find the volume of a prism the area of whose base is 24 square inches and altitude 7 feet. Ans. 168 cubic feet. 29. Find the surface of a cube whose sides are each 11 inches. Ans. 726 square inches. 30. Find the surface of a cubical cistern whose contents are 373,248 cubic inches. Ans. 180 square feet. 31. Find the lateral surface of a right prism whose altitude is 2 feet, and whose base is a regular hexagon of which each side is 10 inches long. 32. Find the depth of a cubical cistern which shall hold 1600 gallons, each gallon being 231 cubic inches. Ans. 5.98 feet. 33. If the dimensions of a rectangular parallelopiped are 20.5 feet, 183 feet, and 6.75 feet, what is the edge of an equivalent cube? Ans. 11.4 feet. 34. Find the depth of a cubical box which shall contain 100 bushels of grain, each bushel holding 2150.42 cubic inches. Ans. 4.9 feet. 35. If the dimensions of a rectangular parallelopiped are 9,16, and 25, what is the edge of an equivalent cube? 36. Find the dimensions of the base of a rectangular parallelopiped whose volume is 60, surface 94, and altitude 3. 37. Find the ratio of two rectaugular parallelopipeds, if their altitudes are each 4 feet, and their bases 6 feet by 3 feet, and 12 feet by 8 feet, respectively. 38. Find the ratio of two rectangular parallelopipeds, if their dimensions are 5, 6, 8, and 10, 12, 16, respectively. 39. Find the volume of a right triangular prism, if the height is 8 inches, and the sides of the base are 6, 5, and 5 inches. 40. Find the lateral area of a right pyramid whose slant height is 4 feet, and whose base is a regular octagon of which each side is 3 feet long. 41. Find the volume of a right quadrangular pyramid whose altitude is 12, and whose base is 4 feet square. 42. Find the volume of a pyramid whose altitude is 20 feet, and whose base is a rectangle 8 feet by %. 43. Find the lateral area of a right pentagonal pyramid whose slant height is 18 inches, and each side of the base 6 inches. Ans. 270 square inches. 44. Find the volume of a pyramid whose altitude is 20 inches, and whose base is a regular hexagon, each side being 6 inches. Ans. 623.5386 cubic inches. 45. Find the lateral area and volume of a right triangular pyramid, the sides of whose bases are 3, 5, and 6, and whose altitude is 6. See (398), Ex. 2. 46. Find the volume of the frustum of a square pyramid, the sides of whose bases are 8 and 6 feet, and whose altitude is 12 feet. Ans. 592 square feet. 47. If a plane be passed parallel to the base of the pyramid in Ex. 44, midway between the vertex and base, find the lateral area and volume of the frustum. 48. The slant height of the frustum of a right pyramid is 6 feet, and the perimeters of the two bases are 18 feet and 12 feet: what is the lateral area of the frustum ? square feet. 49. Find the slant height of the pyramid whose frustum is given in Ex. 48. 50. Find the volume of the frustum of a regular triangular pyramid, the sides of whose bases are 8 and 6, and whose lateral edge is 5. 51. A pyramid 18 feet high has a base containing 169 How far from the vertex must a plane be passed parallel to the base so that the section may contain 81 square feet? 52. The base of a pyramid contains 169 square feet; a plane parallel to the base and 5 feet from the vertex cuts a section containing 81 square feet: find the height of the pyramid. 53. A pyramid 16 feet high has a square base 10 feet on a side. Find the area of a section made by a plane parallel to the base and 6 feet from the vertex. 54. The base of a regular pyramid is a hexagon of which the side is 4 feet. Find the height of the pyramid if the lateral area is eight times the area of the base. 55. Find the total surface of a regular pyramid, (1) when each side of its square base is 12 feet, and the slant height is 24 feet; (3) when each side of its square base is 30 feet, and the perpendicular height is 94 feet; and (3) when each side of its triangular base is 8 feet, and the slant height is 24 feet. 56. Find the volume of a regular pyramid when each side of its square base is 80 feet, and the lateral edge is 202 feet. 57. Find the volume of a regular pyramid whose base is an equilateral triangle inscribed in a circle of radius 40 feet, and whose slant height is 48 feet. 58. Find the lateral edge, lateral area, and volume, (1) of a regular hexagonal pyramid, each side of whose base is 2, and whose altitude is 12; (2) of a frustum of a regular triangular pyramid, the sides of whose bases are 5 V3 and V3, and whose altitude is 3; (3) of a frustum of a quadrangular pyramid, the sides of whose bases are 11 and 1, and whose a altitude is 12; and (4) of a frustum of a regular hexagonal pyramid, the sides of whose bases are 12 and 4, and whose altitude is 12. 59. Find the volume of the frustum of a regular square pyramid, the sides of whose bases are 40 and 16 feet, and whose slant height is 20 feet. 60. The volume of a frustum of a regular hexagonal pyramid is 12 cubic feet, the sides of the bases are 2 and 1 feet: find the height of the frustum. 61. Find the difference between the volume of the frustum of a regular quadrangular pyramid, the sides of whose bases are 4 and 3 feet, and the volume of a prism of the same altitude, whose base is a section of the frustum parallel to its bases and midway between them. 62. Find the dimensions of a cube whose surface shall be numerically equal to its contents. 63. A regular pyramid 8 feet high is transformed into a regular prism with an equivalent base: find the height of the prism. 64. A cube whose edge is 3 feet is transformed into a right prism whose base is a rectangle 3 feet by 2 feet: find the height of the prism. 65. The height of the frustum of a regular quadrangular pyramid is 12 feet, and the sides of its bases are 6 and 10 feet: find the height of an equivalent regular pyramid whose base is 24 feet square. 66. A mound of earth is raised with plane sloping sides and rectangular bases; the dimensions at the bottom are 80 yards by 10, at the top 70 yards by 1, and it is 5 yards high: find its cubical contents. 67. A bath 6 feet deep is excavated; the area of the surface at the top is 100 square yards, at the bottom 81 square yards: find the number of gallons of water it will hold. 68. A railway embankment across a valley has the following measures: width at top 20 feet, at base 45 feet, |