7. The circumference of a circle is 16 chains ; what is the diameter ? Ans. 5.0929+chains. The diameter given, to find the area or content. RULE. Multiply the square of the diameter by the decimal .7854, and the product will be the area. 1. How many square feet are contained in a circle, whose diameter is 4 feet 3 inches? Thus : 4.252=180625 X.7854= 14.1862875 square feet. Ans. 2. What is the value of a circular garden, whose diameter is 6 rods, at the rate of 8 cents per square foot ? Ans. D615.81.6.432. The area af a circle given, to find the diameter. RULE. Divide the area by .7854, and extract the square root of the quotient. 1. The area of a circle is 5 acres, 3 roods, 26 perches, required the diameter. Thus : 946.000-.7854=34.7 poles. Ans. 2. What is the length of a rope fastened to a stake in the centre of a circular field, and the other end to the nose of a horse, which will permit him to feed on 2 acres of land ? Ans. 2.5231 chains. The circumference given, to find the area. RULE. Multiply the square of the circumference by the decimal .07958, and the product will be the area. If the circumference of a circle be 1, the diameter =13.14159=0.31831 ; and the product of this into the circumference is .07958, the area. 1. If the circumference of a circle be 136 feet, what is the area ? Ans. 1472 feet. 2. The circumference of a circle being 37.7, required the area. Thus : 37.72 =1421.29=square, X.07958=113.1062582 =area of the circle. To find the surface of a sphere, globe, or ball. RULE. Square the diameter, and multiply it by the decimal 3.1416, and the product will be the answer. 1. What is the surface of a sphere whose diameter is 12 ? Ans. 122 =144 x 3.1416-=452,3904 2. Required the number of square inches in the surface of a sphere whose diameter is 2 foet, or 24 inches ? Ans. 1809.5616. To find the solidity of a sphere. A sphere or 'globe is a round solid body. RULE, Multiply the surface by the diameter and divide the Sphere. product by 6.; the quotient will be the solidity (multiply the square of the diameter by 3.1416). 1. What is the solidity of a sphere whose diameter is 12 ? Thus : 122=144 X 3.1416=452.3904 x 12=5428.6848-6= 904.7808, Ans. Or, multiply the cube of the diameter by .5236, thus: 12 x 12X12 x.5236=904.7808. 2. What is the solidity of the earth, its mean diameter being 7918.7 miles ? Ans. 259992792082.6374908. To find the solidity of a prism. Definition.-A prism is a body with two equal or parallel ends, either square, triangular, or polygonal, and three or more sides, which meet in parallel lines, running from the several angles at one end to those of the other, Prism. RULE. Multiply the area of the base by the altitude, and the product will be the content. 1. What is the content of a prism, each side of the square which forms the base being 15, and the altitude of the prism 20. feet? Ans. 152=225 x 20=4500 feet. 2. The side of a stick of timber is hewn 3 square, is 10 inches, and the length is 12 feet; required the content. Ans. 10 X 4.33=} perp.=43.3 area at the end, x 12 feet length, =519.6=144=3.6+ content. 3. Required the solidity of a triangular prism whose height is 10 feet, and area of the base 350 feet. Ans. 3500 feet To find the convex surface of a cylinder, Definition.--A cylinder is a round body, whose bases are eircles, like a round column, or stick of timber, of equal bigness at both ends. Cylinder. RULE. Multiply the circumference of its base by the altitude. i. What is the convex surface of a cylinder, the diameter of whose base is 20, and the altitude 50 feet? Ans. 3.1416 X 20 X 50 ==3141.6000 2. Required the convex surface of a cylinder, the circumfer one or whose base is 6509, and altitude 27. Ans. 175743 To find the solidity of a cylinder. RULE. Multiply the square of the diameter of the end by 7854, which will give the area of the base; then multiply the area o? the base by the length, and the product will be the content. 1. Required the solid content of a round stick of timber of equal bigness at both ends, whose diameter is 1.5 and length 20 feet. Thus: 1.5 x 1.5=2.25 X.7854 x 20=35.3430. Ans. Or, 18x18=324 x .7854 x 20=5089.3920--144=35.3430. Ans. 2. What is the solidity of a cylinder, the diameter of whose base is 12, and the altitude 30 ? Ans. 3392.928. 3. How many solid feet in a round stick of timber 16 feet long, and the diameter at each end 15 inches? Ans. 19.635 solid feet. 4. Required the solidity of a cylinder, the diameter of whose base is 30 inches, and the height 50 inches ? Ans. 20.4531 solid feet To find the solidity of a cone. Definition. A cone is a round solid body of a true taper from the base to a point, which is called the vertex. Multiply the area of the base by the altitude, and divido the product hy 3; that is, square the diameter and multiply it by .7854, which gives the area of the base; then multiply kiv the altitude and by 3. Or, the square of the circumference of the base x by .07958, and that product by of the perpen dicular altitude, and the product will be the 'solidity. 1. Required the solidity of a cone, the diameter of whree base is 5 and the altitude 10. Ans, 52= 25 X.7854=19.635 x 10:3=65.45 2. What is the solidity of a cone whose altitude is 27 feet and the diameter of the base 10 feet? Ans. 706.86 3. Required the solidity of a cone, the diameter of whese base is 18 inches, and its altitude 15 feet. Ans. 8.83575 feet. 4. If the circumference of the base of a cone be 40 feet, and the height 50 feet, what is the solidity? Ans. 2122.1333 feet 5. The top of a cistern is 5.5 feet, the bottom 4.75 feet, and the height, or depth, 7.25; how many hogsheads will it con tain ? Ans. 17.77. RULE. To find the solidity of a pyramid. Definition.- A pyramid is a solid, whose sides are all triangles, meeting in a point at the vertex, and the base any plane figure whatever. Pyram. Multiply the area of the base by the altitude, and divide the product by 3. 1. Required the solidity of a pyramid, of which the area of the base is 95 and the altitude 15. Ans. 95 X15=1425:3=475. 2. What is the solidity of a pyramid, the area of whose base is 403 and the altitude 30 ? Ans. 4030. 3. Required the solidity of a triangular pyramid, whose height is 30, and each side of the base 3 ? Thus: 32 x.433013=3.897117 area of the base, then 3.897117 x=38.97117=solidity required. 4. A pyramid with a square base, of which each side is 30, has an altitude of 20; what is the solid content ? Ans. 6000. To measure a parallelopipedon. Definition.--A parallelopipedon is a solid of three dimensions, length, breadth, and thickness; as a piece of timber exactly square, whose length Parallelopipedon, is more than the breadth and thickness; the ends are called bases, which are equal. The solidity of a parallelopiped is equal to the product of the base into the perpendicular altitude. And a parallelopiped and a cylinder, which have equal bases and altitudes, are equal to each other. . Find the area of the base, then multiply that by the length, and it will give the solid content. 1. If the side of a stick of timber is 1.75 feet, and the length 9.5 feet, to find the content. Thus: 1.752=3 0625 =area of base x 9.5 =-29.09375 content. Ans. 2. A vessel 3.5 feet each side within, and 5 feet deep, to find the content. Ans. 3.5 X 3.5=12.25 X 5 =61.25 content. 3. A piece of timber is 1 foot 6 inches broad, 9 inches thick. 9 feet 6 inches long; required the content. Ans. 1.5 X.75=1.125 X 9.5=10.6875 content. 4. What is the solidity of a cylinder whose height is 121 and diameter 45.2 ? Ans. 45.22 X.7854 x 121=192442.6. 5. The Winchester bushel is a hollow cylinder 18 inches in diameter, and 8 inches deep; what is its capacity? Ans. 18.52 X.7853982 - 268.8025 X 8 -- 2150.42 cubic inches RULE. 6. There is a cistern under ground, in the form of a parallel ogram, in length 16 feet, width 12 feet, depth 9 feet; required the number of hogsheads it will contain. Thus : 16 x 12 x9= 1728 cubic feet of space, then 17282=2985984 cubical inches, which divide by 231, the number of inches in a gallon=12926 gallons 63=205 hogsheads, 1131 gallons. Ans. 7. How many hogsheads will a circular cistern contain which is 7 feet in diameter, and 9 feet in depth ? Ans. 41 hogsheads, 7 gallons, 3 quarts, 1 pint. When the breadth and thickness of a piece of timber are given in inches, to find how much in length will make a solid foot. RULE, Divide 1728 by the product of the breadth and depth, and the quotient will be the answer or length, making a solid foot. In a piece of timber 11 inches broad and 8 inches thick, how many inches in length will make a solid foot ? Ans. 11 x8=88)1728(19.6. To find how many solid feet a round stick of timber of equal thick ness will contain, when hewn square. RULE. Multiply twice the square of its radius, in inches by the length in feet, then divide the product by 144, and the quotient will be the answer. 1. Admitting the radius of a round stick of timber to be 11 inches, and its length 20 feet, how many solid feet will it contain when hewn square ? Thus : 11x11 X2 X 20=4840-144 =33.6 solid feet of hewn timber. To find how many feet of square-edged boards of a given thick ness, can be sawn from a log of a given diameter. RULE. Find the solid content of the log when made square by the last rule; then say, as the thickness of the board, including the saw-calf, is to the solid foot :: so is 12 inches to the number of feet of boards. 1. How many feet of square-edged boards 1.25 inches thick, including the saw-calf, can be sawn from a log 20 feet long and 24 inches in diameter ? Thus: 12 x 12 x2=288X20=5760 • 144=40 feet solid content; then, 1.25 : 40 :: 12 : 384 feel of boards, Ans. |