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466. The Applications of Measures to the farm, the bousehold, the mechanic arts, etc., are so extensive that we now present a distinct treatment of the subject.
467. These Practical Measurements include Measures of Surface, Measures of Volume, and Measures of Capacity.
MEASURES OF SURFACE. 468. A Surface is that which has length and breadth without thickness.
469. The Area of a surface is expressed by the number of times it contains some other surface used as a unit of measure.
THE RECTANGLE. 470. A Rectangle is a plane surface having four sides and four right angles. A slate, a door, the sides of a room, etc., are examples of rectangles.
471. A Rectangle has two dimensions, length and breadth. A Square is a rectangle in which the sides are all equal.
472. The Area of a rectangle is the surface included within its sides. It is expressed by the number of times it contains a small square as a unit of measure.
Rule I. – To find the area of a square or rectangle, multiply its length by its breadth.
For, in the rectangle above, the whole number of little squares is equal to the number in each row multiplied by the number of rows, which is equal to the number of linear units in the length multiplied by the number in the breadth.
Rule II.- To find either side of a square or rectangle, divide the area by the other side.
NOTES.-1. The sides multiplied must be of the same denomination, and the product will be square units of that denomination, which may be reduced, if necessary, to higher denominations.
2. In dividing, the linear unit of the side must be of the same name as the square unit of the area, and the quotient will be linear units of the same denomination.
EXAMPLES FOR PRACTICE. 1. What is the area of a rectangular lot 150 ft. long and 80 ft. wide ?
SOLUTION.—To find the area, we multiply the length by the breadth, and we have 150 X 80 = 12000 sq. ft.; reducing this to square yards, we have 1333} sq. yd.
2. How many square yards in the surface of a blackboard 24 ft. long by 41 ft. wide ?
Ans. 12 sq. yd. 3. A room 18 feet wide bas a floor containing 360 sq. ft; what is its length ?
Ans. 20 ft. 4. How many square feet in the walls of a room 25 ft. long, 17 ft. 6 in. wide, and 9 ft. 6 in. high? Ans. 8071 sq. ft.
5. What is the surface of a cubical box, each of whose dimensions is 2 feet 9 in.?
Ans. 453 sq. ft. 6. How many sq. feet in the surface of a chest 3 ft. 9 in. long, 2 ft. 6 in. wide, and 1 ft. 9 in. high? Ans. 40 sq. ft.
7. A lady wishes to set out tulips in a bed 12 ft. long and 3 ft. wide. How many can be planted at a distance of 9 in. apart and 41 in. from the edge ? Ans. 64 tulips.
8. A garden 160 ft. long and 105 ft. wide has a walk around it 7 ft. in breadth; how much ground is contained in the walk?
Ans. 3514 sq. ft.
THE TRIANGLE. 473. A Triangle is a plane surface having three sides and three angles; as ABC.
474. The Base is the side upon AZ wbich it seems to stand; as, AB. The Altitude is a line perpendicular to the base, drawn from the angle opposite; as, CD.
475. A triangle which has its three sides equal is called equilateral ; when two sides are equal it is called isosceles ; when its sides are unequal it is called scalene.
Rule 1.— To find the area of a triangle, multiply the base by one-half of the altitude.
Rule II.— To find the base or altitude of a triangle, divide the area by one-half of the other dimension.
EXAMPLES FOR PRACTICE. 1. What is the area of a triangle whose base is 15 ft. 6 in. and altitude 8 ft. 9 in. ?
SOLUTION.—To find the area, we multiply the base by one-half the altitude; 15.X 43= 671. sq. ft., or 67 sq. ft. 117 sq. in.
2. How many square yards in a triangle whose base is 20 ft. 9 in., and altitude 10 ft. 11 in. ?
Ans. 12 sq. yd. 5 sq. ft. 371 sq. in. 3. Required the area of the gable end of a house 32.5 ft. wide, the ridge being 15.25 feet above the wall.
Ans. 27 sq. yd. 4 sq. ft. 117 sq. in. 4. A triangular lot contains 233 sq. yd. 6 sq. ft. 108 sq.in.; its base is 165 ft.; what is its altitude ? Ans. 25 ft. 6 in.
5. I have a triangular flower-bed containing 48 sq. ft. 63 sq. in., whose altitude is 6 ft. 3 in.; what is the base ?
Ans. 15 ft. 6 in. 6. The gable end of a house contains 47 sq. yd. 6 sq. ft., the width of the house being 17 yd. 1 ft.; what is the height of the ridge ?
Ans. 16 ft. 6 in.
THE CIRCLE. 476. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within, called the centre.
477. The Circumference of a circle is the bounding line; any part of the circumference, as BC, is an Arc. An arc of one-fourth of the circumference is called a Quadrant.
478. The Diameter is a line passing through the centre and terminating in the circumference; as, AB. The Radius is a line drawn from the centre to the circumference; as, OD.
Rule I.– To find the circumference of a circle, multiply the diameter by 3.1416.
Rule 11.— To find the diameter of a circle, multiply the circumference by .3183.
Rule III.— To find the area of a circle, multiply the circumference by one-fourth of the diameter, or multiply the square of the radius by 3.1416.
EXAMPLES FOR PRACTICE. 1. The diameter of a circle is 15 ft. 9 in.; what is its circumference ?
SOLUTION.—To find the circumference, we multiply the diameter by 3.1416; 3.1416 X 15% = 49.4802; hence the circumference equals 49.4802 ft.
2. What is the length of the tire of a carriage wheel 4 ft. 6 in. in diameter?
Ans. 14.1372 ft. 3. In a square in a certain city is a fountain whose basin is 4 ch. 5 li. in circumference; what is its diameter?
Ans. 1 ch. 28.9115 li. 4. The end of the minute-hand of a church clock passes over 25 inches in 15 minutes; what is the length of the minute-hand?
Ans. 31.83 in. 5. How much ground is occupied by a circular lighthouse, its circumference being 50 ft.? Ans. 198.9375 sq. ft.
6. Within a circular plot 50 rods in diameter is a circular pond, whose edge is everywhere 6 rods from the edge of the plot; what is the area of the pond ? Ans. 1134.1176 P.
7. A walk 3 ft. wide extends around the above mentioned plot; what is the area of the walk? Ans. 7803. 7344 ft.
MEASUREMENT OF LAND. 479. The Unit of Measure of land is the Acre, which is sometimes divided into square rods and sometimes into square chains. Hundredths of an acre are also frequently used.
In 1802, Col. Jared Mansfield, Surveyor-General of the North-Western
Territories, adopted a convenient method of laying out Government lands. The country was divided by parallels and meridians 6 miles apart, into squares containing 36 square miles, called Tounships. The townships are divided into square miles, called Sections, and each section into quarter-sections. Hence, 640 acres make a section, and 160 acres a quarter-section. The quarter-sections are still further subdivided into half-quarter-sections, quarter-quarter-sections, and lots. Lots are often of irregular form on account of natural boundaries, but contain, as near as may be, a quarter-quarter-section.
Å Township in the newer States, laid out as explained above, is a square of 36 miles; but in the older States the townships are irregular in shape and variable in size. A Township is a division of a county made for convenience in holding elections. There must be at least one place of voting in each township.
NOTE.—The pupil will remember that rods multiplied by rods give square rods, Chains by chains give square chains; also, that 1 acre=10 square chains or 160 square rods.
EXAMPLES FOR PRACTICE. 1. I have a rectangular lot 40 ch. long and 36 ch. wide; · bow many acres does it contain ?
Solution.—The area equals 40x36, or 1440 sq. ch. which, reduced to acres by dividing by 10, equal 144 acres.
2. A has a square lot 32 chains on a side; how many acres does it contain ?
Ans. 102 A. 64 P. 3. A rectangular lot contains 144 A. 135 P.; the length of one side is 43 ch.; what is the other? Ans. 33.68+ ch.
4. The length of a field is 76 rd. 10 ft. 5 in, and breadth 44 rd. 7 ft. 9 in.; what is the area ?
Ans. 21 A. 47 P. 209 sq. ft. 141 sq. in. 5. I wish to fence a quarter-section with hemlock rails 84 ft. long, lapping 6 inches, the fence being 6 rails high ; how many rails will be required, and what will be the cost at $42 p M. ?
Ans. 7920 rails; $332.64. 6. A field 16 chains long contains 16 acres, while another field of the same width contains only 12 acres; what is the length in rods of the latter field ?
Ans. 48 rods. 7. How much less will it cost to fence a field 64 rods square than a rectangular field 21 times as long and fas wide, if fencing cost $2.75 a rod?
Ans. $316.80. 8. In a piece of ground 64 rods square, I planted 4 acres of corn, 350 square rods with potatoes, 20 rods square with vegetables, 6 acres with raspberries and blackberries, and