Er. Suppose the greater end DC 24, the less end AB 20, and their distance EF 5, required the area ABCD. Def. As a line can only be measured by a line, and a surface by a surface, so a solid can only be measured by a solid; therefore solid measure is the finding the number of cubic inches, feet, yards, &c. contained in any thing that consists in length, breadth, and thickness. Prob. 25. To find the area of a prism. Multiply the area of the base or end, by the perpendicular height, and the product will give the solidity. Er. What is the solidity of a cube whose side is 12 inches? 12x12=144; the area of the end, or 144 inches at one inch deep. 144 X 12=1728, the number of cubic inches in a cubic foot. Ex. 2. What is the solidity of a parallelopiped, whose length is 10 feet the breadth 5f. 9i. and the depth 3f. 6i. ? f. i. 5 9 3 6i 2 10 6 17 3 20 1 6 area of the base. 10 3 O answer. Ex. 3. Required the area of a triangular prism, the height being 12f. 3in. one side of the base 3f. 6in. and the perpendicular of the triangle to that side 2f. 4in. Er. 4. What is the solidity of a cylinder, whose height is 12 feet, and the diameter of the base 2.5 feet? (25) x 7854 x 12=58 905000 the solidity required. The work at length. 2.5 2.5 125 50 6:25 78.5 2500 3125 5000 4375 4.908750 area of the base. 58'905000 Ex. 5. What is the solidity of a cylinder, when the circumference of the base is 7.85 feet, and the height 12 feet? (7.85)*x 0758x12=58 847022600 the solidity of the cylinder. Prob. 16. To find the solidity of a pyramid. Multiply the area of a base or end by the perpendicular height, and one-third of the product will give the solidity. Ex. 1. What is the solidity of a square pyramid, of which the height is 9f. 6in. and each side of the base 2f. 3in.? Ex. 2. Required the solidity of a cone, the diameter of the base being 2f. 6in. and the height 12f. f. in. 2 6=25 2.5 x 2'5 × 7854 x 12=58'905 the solidity of a cylinder of the same base and altitude. Prob. 17. To measure the frustrum of a square pyramid. To the rectangle of the sides of the two ends, add the sum of their squares; that sum being multiplied by the height, one-third of the product will give the solidity. Er. In the frustrum of a square pyramid, one side of the greater base being 3f. 6in.; each side of the lesser end or top 2f. 3in.; and the perpendicular height 6f. 9in.; required the solidity. f. in. 56.671875 solidity of the frustrum. Method 2. To the rectangle of the sides of the two bases, add onethird of the square of their difference; that sum being multiplied by the height, will give the solidity. Ex. In the frustrum of a square pyramid, let one side of the greater base be 3f. 6in.; each side of the top 2f. 3in.; and the perpendicular height 6f. 9in.; required the solidity. f. in. 3 6 f. in. Prob. 18. To measure the frustrum of a cone.. To the rectangle of the two diameters, add the sum of the squares of these diameters; multiply the sum by 7854, and that product by the length; then one-third of the last product will give the solidity. Note. If the circumferences are given, proceed in the same manner, only multiply by 07958, instead of 7854. Er. What is the solidity of the frustrum of a' cone, the diameter of the greater end being 3 feet, and that of the lesser end 2 feet, and the atitude 9 feet? 44.7678 the solidity of the frustrum. Def. A PRISMOID is a solid contained under six planes, the ends being parallel, but unlike rectangles; and the other four sides, each opposite, two are equal trapeziums. Prob. 19. To measure a prismoid. Multiply the length at the greater end by the breadth at the lesser end, and the length at the lesser end by the breadth at the greater end, To half the sum of the two products, add the areas of the two ends; that sum multiplied by one-third of the height, gives the solidity. Ex. What is the solidity of a prismoid, whose greater end is 12 inches by 8, and the lesser end 8 inches by 6, and the length or height 5 feet? 12×6=72 2)136 68 half the sum of the products. 212 sum. 20 one-third of the height. 1728)4240(2 solid ft. and 784 solid in. the Ans, 3456 |