To find the area of a circle, the diameter being given. RULE.-Multiply the square of the diameter by ,7854, and the product will be the area. Or multiply half the diameter by half the circumference, and the product will be the area. 12 Ө EXAMPLES. 1. What is the area of that circle, whose diameter is 12 rods? ,7854-113,0976 rods. given circle. 12x12=144x The area of the DEM-We first multiply the diameter by itself, which gives the area of a square whose side is 12 rods, because multiplying the side of a square by itself gives the area of a square whose side is the given number; the area of a square is one foot, one yard, &c., when its side is 1; but the area of a circle whose diameter is 1, is only ,7854, therefore it is evident that we must multiply the square of the diameter by ,7854, or by ;785398, which is nearer the truth, for the area. 2. What is the area of a circle whose diameter is 14 rods? Ans. 153,9384 square rods. To find the circumference of a circle, the diameter being given: or the circumference given, to find the diameter. RULE.-Multiply the diameter by 3,141592, and the product will be the circumference. Or, as 7: is to 22:: so is the diameter: to the circumference. Or, as 22: is to 7:: so is the circumference: to the diameter; or, more exactly, as 113: is to 355:: so is the diameter: to the circumference, or, as 355: is to 113 :: so is the circumference: to the di ameter. EXAMPLES. 1. What is the circumference of a circle whose diameter is 30 rods? As 113 355:30: Ans. 94,248 rods, nearly. 2. What is the diameter of a circle whose circumference is 94,248 rods? As 355 113:: 94,248: Ans. 30 rods. : The area of a circle being given, to find the diameter. RULE. Divide the area by ,7854, and extract the square root of the quotient; the root will be the diameter sought. Or, as 355 452 :: so is the area: to the square of the diameter; or, as 1: 1,273239 :: so is the area to the square of the diameter; or you may multiply the square root of the area by 1,12837, and the product will be the di ameter. : EXAMPLES. 1. What is the diameter of that circle whose area is 1 acre or 160 square rods? Ans. 14,273 rods, nearly. 160 rods+,7854-v203,7178-14,273 nearly. 2. What length of rope may be tied to a horse's head, and the other end to a stake, to give him the liberty of eating two acres of grass? Ans. 55 yards. NOTE.-The area of circles are to each other, as the squares of their diameters. To find the area of a globe or ball. RULE.-Multiply the whole circumference by the whole diameter, and the product will be the area. NOTE. The area of a globe or ball is 4 times as much as the area of the circle of the same diameter, hence the rule is obvious. EXAMPLES. 1. What is the number of square miles on the surface of the earth, allowing the diameter to be 7911 miles, and the circumference 24853 miles? Ans. 196612083 sq. miles. 2. Suppose the ball on the top of St. Paul's Church is 6 feet in diameter; what did the gilding of it cost, at 34d. per square inch? Ans. £237 10s. 1d. To find the area of a circle, the circumference and diameter being given. RULE.-Multiply half the circumference by half the diameter, and the product will be the area. Or multiply the whole circumference by the whole diameter, and one fourth of the product will be the area, EXAMPLES. 1: What is the area of a circle, whose diameter is 7, and circumference 22? 11x3=381, or 22×7=154÷4=381. Ans. 382. 2. What is the area of a circle, the diameter of which is 10 feet 6 inches, and the circumference 31 feet 6 inches? Ans. 82 feet 8 inches. MENSURATION OF SOLIDS, Teaches to find the solidity of bodies that have length, breadth, and thickness. DEFINITION.-Mensuration signifies to measure, hence measuring urfaces is called mensuration of superficies; and measuring solids is alled mensuration of solids, that is, to measure a solid, so as to express ts content in solid or cubick inches, feet, yards, &c. To find the solidity of a cube. RULE.-Multiply the side of the cube by itself, and that product again by the side, the last product will be the solidity. feet? EXAMPLES. 1. How many solid or cubick inches, in a cube of marble whose side is 24 inches? 24×24×24=Ans. 13824 sol. in. 2. What is the solidity of a cube, the side of which is 5 Ans. 125 solid feet. To find the solidity of a parallelopipedon, that is, a solid contained by six quadrilateral planes, every opposite two of which are equal and parallel. RULE.-Multiply the length by the breadth, and that product again by the thickness or height, and it will give the solidity. EXAMPLES. 1. What is the solidity of a parallelopipedon, whose length is 12 feet, breadth 4 feet, and height 6 feet? 12x4x6-Ans. 288 solid feet. 2. How many solid feet in a load of wood 8 feet long, 34 feet wide, and 34 feet in height? Ans. 98 feet. 3. What number of bricks 8 inches long, 4 inches wide, and 2 inches thick, will it require to build a house 46 feet long, 38 feet wide, and 20 feet high, and the walls to be 1 foot thick? Ans. 88560 bricks. To find the solidity of a cylinder. DEFINITION. A cylinder is a round body whose bases are circles, like a round column or stick of timber, of equal bigness from end to end. RULE.-Multiply the area of the base by the perpendicular height, and the product will be the solidity. . EXAMPLES. 1. What is the solidity of a cylinder, the height of which is 5 feet, and the diameter of the end 2 feet? Ans. 15,708 ft. 2. One evening I chanc'd with a Tinker to sit, Whose tongue ran a great deal too fast for his wit; He talk'd of his art with abundance of mettle; So I ask'd him to make me a flat-bottomed kettle; Let the top and the bottom diameters be In just such proportion as five are to three; Twelve inches the depth I propos'd, and no more; And to hold in ale gallons, seven less than a score. He promis'd to do it, and straight to work went; But when he had done it he found it too scant. He alter'd it then, but too big he had made it; For thougn it held right, the diameters fail'd it; Thus altering it often too big and too little, Ans. (14,64 inches, bottom diameter. 24,4 inches, top diameter. NOTE. The kettle is not a cylinder, the top and bottom diameters being unequal, yet there is sufficient given in this and the preceding rules for finding the diameters. To measure a Sphere or Globe. DEFINITION.-A sphere or globe is a round, solid body, in the middle of which is a point, from which all lines drawn to the surface are equal. RULE.-Multiply the cube of the given diameter by ,5236, and the product will be the solid contents. EXAMPLES. ard of 1. The diameter of a globe is 12 inches; how many cubick or solid inches does it contain? 12×12×12=1728X,5236 equal to Ans. 904,7808 solid inches. DEM. By cubing the diameter, the product, 1728, is the solidity of a cube whose side is 12 inches, we then multiply by the decimal, ,5236, because ,5236 is the solidity of a globe whose diameter is 1. NOTE-A cube whose side is one inch, contains one cubick or solid inch. A globe whose diameter is one inch, contains ,5236 of an inch. 2. Suppose the diameter of the earth is 7911 miles; how many solid or cubick miles does it contain? Ans. 259,235,092,532 6316 solid miles. QUESTIONS ON MENSURATION. What is Mensuration of Superficies? A. It teaches how to measure surfaces or area. How do you find the surface or area of a square? A. Multiply the side of the square into itself, and the product will be the area. How do you find the area of a parallelogram or long square? A. Multiply the length by the breadth, and the product will be the area. How do you find the area of a right angled triangle? A. By multiplying the base by one half the perpendicular, the product will be the area. How do you find the area of a circle? A. Multiply the square of the diameter by ,7854, the product will be the area. Why multiply by,7854? A. Because the area of a circle is ,7854 when the diameter is one. How do you find the area of a globe or ball? A. Multiply the whole circumference by the whole diameter, and the product will be the area. What does mensuration of solids teach? A. It teaches how to measure solids. How do you find the solidity of a cube? A. Cube one side, and the product will be the solidity. How would you find the number of cubick feet in a load of, wood that is 8 feet long, 3 feet wide, and 4 feet high? A. Multiply the length, breadth, and height together, the product will be the solidity. How do you find the solidity of a cylinder? A. Multiply the area of the base by the perpendicular height, and the product will be the solidity. How do you find the solidity of a globe? A. Multiply the cube of the diameter by,5236, the product will be the solidity. Why multiply by ,5236 ? A. Because,5236 is the solidity of a globe whose diameter is 1. DUODECIMALS, Is a rule much used by workmen and artificers, in computing the contents of their work. The rule has derived its name from the Latin word duodecim, which signifies twelve. A foot, which is called an integer, is divided duodecimally, that is, into twelve parts, called inches or primes; an inch or priine is divided into twelve parts, called seconds; a second is divided into twelve parts, called thirds, and so on. But dimensions are usually taken in feet, inches, and quarters; the parts smaller than these are generally neglected, being of little or no consequence. RULE.-1st. Set down the two given dimensions, i. e. length and breadth, one under the other, so that feet may stand under feet, inches under inches, &c. 2d. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each directly under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. 3d. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting, however, what is below the parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as there are like parts of a foot in the inches. Then add the products together, as in Compound Addition, carrying 1 to the feet for every 12 inches; the result will be the answer, or area, in square feet and inches. EXAMPLES. 1. How many square feet in a board, 14 feet 9 inches long, and 2 feet 6 inches wide? Ans. 36 feet, 10 inches. |