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 mul tiplied 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. h 3 459. The Solid Contents of a Cylinder = D2X .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 + D2 × .7854 + D × D' X .7854) X (D2 + D'2 + D × D') × .7854 × h h = 3 462. The Convex Surface of a Cylinder D X 3.1416 X h. = s.h. 2 464. The Convex Surface of a Frustum of a Cone= 8. h. (D x 3.1416+ D'× 3.1416) X 2 465. The Convex Surface of a Sphere DX 3.1416 X DD2 X 3.1416. .5236 466. The Solid Contents of a Sphere D2 × 3.1416 * 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 8 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 34 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 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.. 19. What proportion do the cubical contents of a sphere bear to the contents of a cylinder which will just contain it? Ans. 4. 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 POLY 1 in. GONS. 3 in. square. 2 in. square. 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 con. tain 1 square inch, one 2 inches square will contain 4 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 cisern in 3 hours, what must be the diameter of a pipe to empty he same cistern in 12 hours? By Proportion, 1 : 3 = 22 : 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 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. 5 The hypothenuse of a right-angled triangle is 40 feet; what must be the hypothenuse of a similar triangle that it may contain twice the area? Ans. 56.568+. 6. If a circular lot of land which is 10 rods in diameter contains 78.5398 square rods, what number of rods will a lot contain which is 5 rods in diameter ? Ans. 19.63495. 7. The area of a triangle whose base is 24 feet is 120 feet; what is the area of a similar triangle whose base is 96 feet? Ans. 1920 feet. 8. The Winchester bushel is 18 inches in diameter and 8 inches deep; what must be the diameter of a circular measure 6 inches deep, that it may hold a bushel? Ans. 21.36 inches. 9. I have a circular flower-garden, the circumference of which is bordered with 75 yards in length of sodding; how many yards will be required to border a circular garden of the area? Ans. 61.237+ yards. 10. Having a triangular board 7 feet long, what distance from the base end shall I cut it to divide it into two equal parts? Ans. 2.197-ft. SIMILAR SOLIDS. NOTE.Angular solids are similar when their angles are equal each to each, and arranged in the same way, and their corresponding edges proportional. The following proposition may be easily proved by geometry: : 472. The Solidities of Cubes, Spheres, and all Similar Solids are to each other as the cubes of their corresponding dimensions. ILL. EX., I. How many globes of 6 inches diameter can be made from a globe of 48 inches diameter ? By Proportion, 63: 4831 (globe): 512 globes, Ans. ILL. EX., II. If a conical stack of hay which contains of a ton is 6 feet high, what is the height of a similar stack which contains 3 tons ? |