307. To find the area of a trapezoid. (65) RULE.-Multiply half the sum of the two parallel sides by the perpendicular distance between them, and the product will be the area. 308. To find the area of a trapezium, or an irregular polygon. RULE.-Divide it into triangles, and then find the area of these triangles by Art. 306, and add them together. 309. To find the diameter and circumference of a circle, either from the other. (67) RULE 1. As 7 is to 22, so is the diameter to the circumference, and as 22 is to 7, so is the circumference to the diameter. RULE 2.-AS 113 is to 355, so is the diameter to the circumference, and as 355 is to 113, so is the circumference to the di ameter. RULE 3.-As 1 is to 3.1416, so is the diameter to the circumference, and as 3.1416 is to 1, so is the circumference to the diameter. As 113: 355:: 14: 4311, Ans. As 1:3.1416: : 14 : 43.9824, Ans. 2. Supposing the diameter of the earth to be 7958 miles, what is its circumference? Ans. 25000.8528 miles. As 355 113: 50: 15.9155, Ans. As 3.14161: 50: 15.9156, Ans. 4. Supposing the circumference of the earth to be 25000 miles, what is its diameter? Ans. 7957 nearly. 310. To find the area of a circle. RULE.-Multiply half the circumference by half the diameter, or the square of the diameter by .7854,- -or the square of the circumference by .07958,-the product will be the area. 311. The area of a circle given to find the diameter and circumference. RULE 1. Divide the area by .7854, and the square root of the quotient will be the diameter. 2. Divide the area by .07958, and the square root of the quotient will be the circumference. 312. To find the area of an oval, or ellipsis. RULE.-Multiply the longest and shortest diameters together, and the product by 7854; the last product will be the area 313. To find the area, or surface, of a globe or sphere. RULE.-Multiply the circumference by the diameter, and the product will be the area. 1. How many square feet in the surface of a globe whose diameter is 14 inches, and circumference 44? 44×14 616, Ans. 2. How many square miles in the earth's surface, its circumference being 25000, and its diameter 7957 miles ? Ans. 198943750. 3. What is the area of the surface of a cannon shot, whose diameter is 1 inch? Ans. 3.1416 inches. 4. How many square inches in the surface of an 18 inch artificial globe? Ans. 1017.8784. 2. Mensuration of Solids. 314. Mensuration of Solids teaches to determine the spaces included by contiguous surfaces, and the sum of the measures of these including surfaces is the whole surface of the body. The measure of a solid is called its solidity, capacity, content, or volume. The content is estimated by the number of cubes, whose sides are inches, or feet, or yards, &c. contained in the body. 315. To find the solidity of a cube. (254) RULE.-Cube one of its sides, that is, multiply the side by itself, and that product by the side again, and the last product will be the answer. 1. If the length of the side of a cube be 22 feet, what is its solidity? 22X22X22-10648, Ans. 2. How many cubic inches in a cube whose side is 24 inches? Ans. 13824. 316. To find the solidity of a parallelopipedon. (69) RULE.-Multiply the length by the breadth, and that product by the depth; the last product will be the answer. 1. What is the content of a parallelopipedon whose length is 6 feet, its breadth 2 feet, and its depth 13 feet? 6X2.5X1.75 26.25, or 26 feet. 2. How many feet in a stick of hewn timber 30 feet long, 9 inches broad, and 6 inches thick? Ans. 11 feet. 317. To find the side of the largest stick of timber that can be hewn from a round log. RULE.-Extract the square root of twice the square of the semidiameter at the smallest end for the side of the stick when squared. 1. The diameter of a round log at its smallest end is 16 inches; what will be the side of the largest squared stick of timber that can be hewed from it? 8X8X2=11.31 in. Ans. 2. The diameter at the smallest end being 24 inches, how large square will the stick of timber hew? Ans. 16.97 in. 318. To find the solidity of a prism, or cylinder. RULE.-Multiply the area of the end by the length of the prism, for the content. 1. What is the content of a triangular prism, the area of whose end is 2.7 feet, and whose length is 12 feet? 2.7X12 32.4 ft. Ans. 2. What number of cubic feet in a round stick of timber whose diameter is 18 inches, and length 20 feet? Ans. 35.343. 319. To find the solidity of a pyramid, or cone. RULE.-Multiply the area of the base by the height, and one third of the product will be the content. 1. What is the content of a cone whose height is 12 feet, and the diameter of the base 2 feet? 2+21=1=25-6, and 6.7854× 12÷3 20.453125, Ans. triangular 2. What is the content of a pyramid, its height feet, and the sides being 5, 6 and 7 Ans. 71.035+. of its base feet? |