451. The Solidity of a Prism or of a Cylinder equals the area of its base multiplied by its height; for it is evident that a prism or cylinder 1 inch high must contain as many cubic inches as there are square inches in the base; and if it is 2, 3, or any number of inches high, it must contain 2, 3, or that number of times as many solid inches. 452. The Convex Surface of an Upright Prism or Cylinder equals the perimeter of one of its bases multiplied by its height; for it is evident that, if the prism or cylinder is 1 inch high, its convex surface contains as many sq. inches as there are inches in the perimeter; and if the prism or cylinder is any number of times 1 inch in height, its convex surface must contain that number of times as many square inches. 453. The Solidity of a Pyramid or Cone equals the area of its base multiplied by of its height; for it can be proved that these solids are each of a prism or cylinder of the same base and height. 454. The Convex Surface of a Pyramid or Cone equals the perimeter of its base multiplied by of the slant height; for the convex surface of each may be regarded as composed of triangles whose bases form the perimeter of the base of the solid, and whose height is the slant height of the solid. 455. The Solidity of the Frustum of a Pyramid or Cone equals that of three pyramids or cones whose bases are the upper and lower bases of the frustum and a mean proportional (Art. 373) between the two, and whose height is the height of the frustum. Hence, the solidity equals the sum of the two bases plus the square root of their product, multiplied by of the height of the frus tum. 456. The Convex Surface of the Frustum of a Pyramid or Cone equals the sum of the perimeters of the two bases multiplied by the slant height; for the convex surface of each may be regarded as made up of trapezoids whose parallel sides form the perimeters of the bases, and whose height is the slant height of the frustum. 457. Geometricians have proved that the Convex Surface of a Sphere equals the circumference multiplied by the diameter, or equals the area of four great circles* of the sphere. 458. The Solidity of a Sphere is equal to its surface multiplied by of the radius, or of the diameter, for the sphere may be regarded as made up of pyramids whose bases comprise the surface of the sphere, and whose vertices are at the centre. From the preceding explanations, and by the use of the well established fact that the circumference of every circle is 3.1416 times the diameter, the following formulas for finding the solid contents and convex surfaces of cylinders, cones, frustums of cones, and spheres, are obtained. To save space, D will be used for diameter of lower base, D' for diameter of upper base, h. for height, and s. h. for slant height. 3 459. The Solid Contents of a Cylinder = D2 × .7854×h. 460. The Solid Contents of a Cone D2 X .7854 X. 461. The Solid Contents of a Frustum of a Cone= (D2 × .7854 + D'2 × .7854 + D × D' × .7854) X (D2 + D'2 + D × D') X .7854 ×— h 462. The Convex Surface of a Cylinder = DX 3.1416 X h. × 2 s.h. 465. The Convex Surface of a Sphere =DX 3.1416 × DD2 X 3.1416. .5236 466. The Solid Contents of a Sphere D2 × 3.1416 D XD3 X .5236. A great circle of a sphere is a circle which divides the spnere inte two equal parts. 467. EXAMPLES. 1. How many cubic feet does a block of granite contain, that is 12 feet long, 4 feet wide, and 14 feet thick? Ans. 72 cu. feet. 2. What number of cubic feet are there in a cube whose edge is 1 foot, 11 inches? Ans. 7.041+ cu. feet. 3. How many cubic feet in a prism whose base is a parallelogram 15 feet long and 4 feet wide, and whose height is 9 inches? Ans. 45 feet. 4. Required the contents of a prism whose base contains 81 square yards, and the square of whose height equals 3 times the number of square feet in the base. Ans. 41 cu. yards. 5. Required the contents of a pyramid whose base is the same as the above, and whose height is 5 feet. Ans. 4 cu. yards, 17 cu. feet. 6. Required the contents of a pyramid whose base is 7 feet square, and whose height equals the diagonal of the base. Ans. 161.69 cu. feet. 7. Required the contents of the frustum of a pyramid whose bases are 12 and 108 square feet, and whose height is 18 feet. Ans. 936 cu. feet. 8. What is the convex surface of a prism, the perimeter of whose base is 7 yards, 2 feet, and whose height is 5 yards, 1 foot? Ans. 408 sq. yards. 9. Required the number of square feet in the surface of a four-sided pyramidal roof, the length of each side being 20 feet, and the slant height 18 feet. Ans. 720 sq. feet. 10. What would be the square contents of a four-sided pyramidal roof, the length of each side being 48 feet, and the highest point 10 feet above the eaves? Ans. 2496 sq. feet. 11. Required the number of square feet in the sides of an octangular (eight-sided) tower, the length of each side of the base being 2 feet, 9 inches, that of each side of the top 1 foot, 10 inches, and the height of the tower to the roof, measured on the side 12 feet. Ans. 220 sq. feet. 12. Required the capacity of a cylindrical cistern, measuring 6 feet across and 8 feet deep. Ans. 226.195 cu. feet. 13. Required the capacity of a conical pit, measuring 8 feet across and 5 feet from the edge to the deepest part. Ans. 50.2656 cu. feet. 14. How many quarts of water will a circular tin pan contain, that measures across the bottom 11 inches, across the top 14 inches, the slant height being 3 inches? Ans. 6.65+ quarts. 15. How many cubic feet in a ball 5 feet in diameter ? Ans. 65.45 cu. feet. 16. How many square feet in the surface of the ball? Ans. 78.54 sq. feet. 17. How many square inches of leather will cover a ball 4 inches in diameter ? 18. What proportion do the cubic contents of a cone bear to the contents of a cylinder which will just contain it? Ans. 1. 19. What proportion do the cubical contents of a sphere bear to the contents of a cylinder which will just contain it? Ans. . 20* Suppose, when the moon is 238600 miles from the earth,† that its shadow just reaches the earth's surface, how many cubic miles in the shadow, allowing the diameter of the moon to be 2160 miles, and that of the earth to be 8000 miles? Ans. 283,914,786,355.2 cu. miles. RELATIONS OF CIRCLES, SIMILAR TRIANGLES, AND POLYGONS. 3 in. square. 2 in. square. 1 in. 1 sq. in. 4 sq. in. 9 sq. in. 468. It will be apparent, by the annexed diagrams, that a figure 1 inch square will contain 1 square inch, one 2 inches will contain 4 square square inches, one 3 inches square will contain 9 square inches, and thus, generally, that the areas of squares are to each other as the squares of their edges. †The distance is measured from the centre of the earth to the centre of the moon. The same principle applies to circles, triangles, and all figures that are similar to each other;* hence, 469. I. The Areas of Similar Triangles and Polygons are to each other as the squares of their corresponding dimensions. ILL. EX. A triangle whose base is 10 feet has an area of 15 feet; what is the area of a similar triangle whose base is 12 feet? By Proportion, 102: 12215: 21.6 square feet, Ans. 470. II. The Areas of Circles are to each other as the squares of their diameters, semi-diameters, and circumferences. ILL. Ex. If a pipe of 2 inches diameter will empty a cistern in 3 hours, what must be the diameter of a pipe to empty the same cistern in 1 hours? By Proportion, 11: 322; 8, the square of the diameter of the required pipe. 82.828+ inches, Ans. 1. If the pot to a furnace which consumes 60 lbs. of coal a day is 24 inches in diameter, what amount of coal will be consumed in the same time by a furnace whose pot is 15 inches, all other conditions being the same? Ans. 23.437+ lbs. 2. If a rope 3 inches in diameter weighs 20 lbs., what is the diameter of a rope of the same length which weighs 9 lbs.? Ans. 2.012+ in. 3. If a pipe 4 inches in diameter fills a cistern in 20 minutes, 15 seconds, in what time will a pipe that is 24 inches in diameter fill the same cistern? Ans. 51.84 minutes. 4. If it costs $10.50 to cover a roof whose length is 7. feet, what will it cost to cover a similar roof whose length is 21 feet? Ans. $94.50. * Angular figures are similar when their angles are equal, and their corresponding sides proportional; and, conversely, similar figures have their corresponding sides proportional. |