Frustum of a Pyramid or Cone.

624. The volume of the frustum of a cone can be found by supposing the part cut off to be restored, then finding

the volume of the whole cone, the volume of the restored part, and the difference between the two. This difference is the volume of the frustum.

The heights of the two cones are found from the given dimensions of the frustum in this way. Draw (Fig. 55) the axis, AR, and a slant height, AC, of the whole cone, and between them the radii CR and BQ. Draw also the perpendicular BD, parallel and equal to QR. The two right-angled triangles ARC and BDC have the same angle at C and equal angles at A and B. Hence the triangles are similar and their sides are proportional. That is,

FIG. 55.

The first three terms of this proportion are known; for CD is the difference between the given radii of the two bases of the frustum, DB is the given height of the frustum, and CR the given radius of the larger base.

The value of RA found from the proportion is the height of the whole cone. The height of the small cone, QA, is found by subtracting the given height of the frustum RQ from the height of the whole cone.

With the given radii of the two bases and the two heights RA and QA, the volumes of the two cones are found.

625. Illustrative Example. A common tin milk pan, measuring 12 inches in diameter on the bottom, 15 inches in diameter across the top, and 3 inches deep, contains how many cubic inches ? How many quarts?

CDCR-DR = CR-BQ=7.5-6=1.5.

Ans. The pan contains 503.0487 cu. in. or 8.7 qt.

626. The following is a convenient rule for finding the volume of a frustum of a pyramid or cone, but it cannot be explained without more geometry than can be given here.

Find the area of the lower base, the area of the upper base, and the mean proportional between the two. Multiply the sum of these three results by the height of the frustum and divide the product by three.

This rule can be used to verify the results obtained by the method explained above.

627. Examples for Written Work.

70. How many cubic inches are there in a triangular prism, each side of whose base measures 4 inches and whose height is 8 inches?

71. Find also the convex surface and the total surface of

this prism.

72. How many cubic feet are there in a pyramid 2 feet high and having a base 1 foot square?

73. Find also the convex surface of this pyramid.

74. The base of a pyramid is a square, and its faces are four equilateral triangles. Find the convex surface and the volume of such a pyramid, supposing each edge of it to measure 1 foot.

75. A regular hexagonal prism contains how many cubic inches if its height is 6 inches and each side of its base measures 2 inches?

76. Find the entire surface and the volume of a regular hexagonal pyramid, supposing its height to be 6 inches and each side of the base 2 inches.

77. How many square feet are there in the surface of a pyramidal roof covering a house 40 feet square, if the eaves project 2 feet beyond the walls of the house, and the apex of the roof is 16 feet above the level of the eaves?

78. There is a solid bounded by four equilateral triangles. Find the whole surface of such a solid, supposing each edge of it to measure 1 foot.

79. How many cubic inches are there in a right cylinder whose height is 6 inches and the diameter of the base 4 inches?

80. What is the volume of a right cone of the same base and the same height as the cylinder just described?

81. How many gallons of oil can be stored in a cylindrical iron vat 4 feet in diameter and 9 feet deep?

82. How many square feet of sheet iron are there in a piece of stove pipe 6 inches in diameter and 3 feet long, allowing 1 inch for lapping at the joint?

83. A section of cast-iron water pipe 12 feet long is of uniform thickness, 1 inch, throughout. How many cubic feet of material are there in the pipe, supposing the interior diameter to be 24 inches? What does it weigh, cast iron being 74 times as heavy as an equal bulk of water?

84. How many cubic inches are there in a grindstone 4 feet in diam., 4 inches thick, and having a hole at the center 4 inches square? What is the weight of this stone if it weighs 2.45 times as much as an equal bulk of water?

85. How many cubic inches are there in a bushel measure, cylindrical in form, 18.5 in. in diameter and 8 in. deep?

86. A two-quart measure is to be made of tin in cylindrical form, the diameter to be two thirds of the depth. Find the dimensions, one quart containing 57 cubic inches. Let the depth.

87. How many cubic inches are there in a box 12 inches square on the bottom, 16 inches square at the top, and 8 inches deep (interior measurements)? (See Article 626.)

88. A piece of granite 20 feet long, 4 feet square at one end and 2 feet square at the other, contains how many cubic feet? How much does it weigh, a cubic foot of granite weighing 162.5 pounds?

89. Measuring a common water pail, I find the diameter of the bottom to be 83 inches, diameter of the top 11 inches, and depth 8 inches. How many quarts does this pail hold, allowing 57.75 cubic inches to a quart?

90. Measure a common tumbler, tin dish, or other vessel made in the form of the frustum of a cone, and find the contents in cubic inches. Now measure the water which fills the vessel in quarts, and see how nearly the two results agree.

91. How many square inches are there in the surface of a ball 4 inches in diameter?

92. How many cubic inches are there in this ball?

93. Assuming the earth to be a sphere 7912 miles in diameter, how many square miles are there in its surface?

94. Suppose a cube, a cylinder, a sphere, and a cone all to have the same dimensions; that is, the edge of the cube is equal to the height and to the diameter of the cylinder; equal to the diameter of the sphere; and equal to the height and to the diameter of the base of the cone. If the edge of the cube measures 6 inches, what is the volume of each solid?

95. Show that if a cylinder, sphere, and cone have the same dimensions (see last example), their volumes are proportional to the numbers 3, 2, and 1.

96. Find the diameter of a sphere which contains 1000 cubic inches.

97. Find the diameter of a sphere whose surface contains 1000 square inches.

98. Find the volume of a sphere whose surface measures 1000 square inches.

99. Find the surface of a sphere which contains 1000 cubic inches.

100. If the height of a right cylinder is equal to half the diameter of its base, how does the area of its convex surface compare with the united area of its two bases?

101. The volume of a right cone is 1000 cubic inches. Find its height and the diameter of its base, the former being to the latter as 5 to 4.

SIMILAR POLYGONS.

628. Similar polygons are polygons which have the same shape, though they may not have the same size. (Art. 501.) All the corresponding sides or other lines of similar polygons are proportional.

629. The areas of similar figures are thus compared:

In Fig. 56 are represented three similar triangles. The second has its sides twice as long, and the third has its sides three times as long, as the corresponding sides of the first. By dividing each side of the second triangle into two, and each side of the third into three equal parts, and drawing lines, we find that the second triangle is made

FIG. 56.

up of four and the third triangle of nine triangles, each equal to the first. The corresponding sides of these similar triangles are proportional to the numbers one, two, and three, while the areas are proportional to the numbers one, four, and nine, which are the squares of the former numbers.