SECTION VII.

PRACTICAL MEASUREMENTS.

332. The Applications of Measures to the farm, the household, the mechanic arts, etc., are so extensive that we now present a distinct treatment of the subject.

333. These Practical Measurements include Measures of Surface, Measures of Volume, Measures of Capacity, and Comparison of Weights and of Money.

MEASURES OF SURFACE.

334. A Surface is that which has length and breadth without thickness.

THE RECTANGLE.

335. 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.

336. A Rectangle has two dimensions, length and breadth. A Square is a rectangle in which the sides are all equal.

337. 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 came denomination.

WRITTEN EXERCISES.

1. How many square feet in a floor 32 ft. long by 21 ft. wide? how many square yards?

SOLUTION.-To find the area, we multiply the length by the breadth, and we have 32×21=672 sq. ft.; reducing this to square yards, we have 743 sq. yd.

2. How many square yards in the surface of a blackboard 27 ft. long by 4 ft. wide? Ans. 12 sq. yd. 3. How many square yards in a garden 215 ft. lon 109 ft. wide? 4. What is the width of a room 25 feet long, whose foor contains 500 sq. ft.?

Ans. 2603

5. A rectangle contains 466 sq. ft., and 6 in. long; how long is the other side?

6. How many square feet in the sides long, 14 ft. 6 in. wide, and 9 ft. 6 in. high? 7. A certain box is 3 ft. 6 in. long, 2 ft.

Ans. 20 ft. one side is 16 ft Ans. 28 ft. 3 in. of a room 18ft Ans. 617.

3 in. wide, and

1 ft. 4 in. high; how many square feet in its surface? Ans. 31 sq. ft. 12 sq. in.

8. What is the surface of a cubical box, each of whose dimensions is 1 ft. 6 in. ?

Ans. 13 sq. ft.

THE TRIANGLE.

338. A Triangle is a plane surface having three sides and three angles; as, ABC.

B

D

339. The Base is the side upon which it seems to stand, as AB. The Altitude is a line perpendicular to the base, drawn from the angle opposite; as, CD.

340. 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 I.-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 the other dimension.

WRITTEN EXERCISES

1. What is the area of a triangle whose base is 25 inches and altitude 18 inches?

SOLUTION. To find the area, we multiply the base by one-half the altitude; 25x9=225; hence the area is 225 sq. in.

2. How many square feet in a triangle whose base is 18 ft. 6 in. and altitude 9 ft. 9 in.? Ans. 90 sq. ft. 27 sq. in. What is the area of the gable end of a house 29 ft. wide being 12 ft. higher than the top of the wall? Ans. 174 sq. ft.

4. The area of a triangular bed of flowers is 25 sq. ft., and its base 10 ft.; what is the altitude?

Ans. 5 ft.

5. The area of a triangular lot is 250 square yards, and its base is 250 ft.; what is its altitude?

Ans. 18 ft.

6. The area of the gable of a house is 378 sq. ft., the base being 14 yards; what is the height of the ridge?

Ans. 18 ft.

THE CIRCLE.

341. 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.

342. 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.

343. 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 ine diameter by 3.1416.

Rule II.-To find the diameter of a circle, multiply the circumference by .3183.

Rule III. To find the area of a circle, multiply the cir cumference by one-fourth of the diameter, or multiply the square of the radius by 3.1416.

WRITTEN EXERCISES.

1. The diameter of a circle is 12 feet; what is its circumference?

SOLUTION. To find the circumference, we multiply the diameter by 3.1416; 3.1416×12 equals 39.27; hence the circumference equals 39.27 ft.

2. What is the circumference of the planet Venus, its diameter being about 7800 miles? Ans. 24504.48 miles. 3. The distance round a circular pond is 500 feet what is the distance across the pond? Ans. 159.15 ft.

4. How many times will a carriage wheel 4 ft. 6 in. in circumference revolve in driving 10 miles? Ans. 11733.

5. I have a circular flower-bed 50 feet in circumference; what is the area of the bed? Ans. 198 sq. ft. 135 sq. in. 6. A cow is fastened to a stake by a rope 16 feet long; what space can she graze over? Ans. 89.36+ sq. yd.

SUPPLEMENTARY PROBLEMS.

To be omitted unless otherwise directed.

7. If the equatorial diameter of the earth is is 7925.75 miles, what are its circumference and the length of a degree of longitude at the equator? Ans. 24899.536+ miles; 69.16+miles. 8. A circular flower-bed being divided into four equal parts by lines drawn from the centre, one section was planted with tulips; what was the area of the tulip-bed, its outer edge being 7 feet? Ans. 15.5967 sq. ft.

MEASUREMENT OF LAND.

344. 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.

Government lands are divided by parallels and meridians into townships, which contain 36 square miles or sections, and each section is sul

divided 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.

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.

WRITTEN EXERCISES.

1. How many square rods in a grass plat 65 ft. long and 15 ft. wide?

SOLUTION.-The area equals 65× 15, or 975 sq. ft.; reducing to square rods, we have 31 sq. rd.

2. How many acres in a rectangular meadow 725' rods long and 400 rods wide? Ans. 1812 A. 80 P. 3. What is the value of a farm 208.7 rods long and 120 rods wide, at $81 an acre? Ans. $12795.917. 4. Mr. A bought 64 A. 116 P. of land for $3.50 per square rod, and sold it for $3.75 per square rod; what did he gain? Ans. $2589. 5. A rectangular pond is 200 rd. 17 yd. long, and 150 rd. 15 yd. wide; required its area.

Ans. 193 A.

6. I have a field 16.5 ch. long and much land does it contain?

137 P. 15 sq. yd. 9.75 ch. wide; how

Ans. 16 A. 14 P.

7. Mr. Wilson's farm contains 163 A. 3 ch., and its lengt is 71 ch.; how many rods of fence would be required to surround it? Ans. 752 rd.

SUPPLEMENTARY PROBLEMS.

To be omitted unless othermise directed.

8. If a township is equally divided among 480 families, how many acres does each family receive, and what part of a section does each receive? Ans. 48 acres; of a section. 9. How many rails are required to fence a quarter-quarter-seotion, the fence being 5 rails high, and each rail 8 ft. long; and what will be the cost at $35 per thousand rails? Ans. 3300 rails; $115.50. 10. A field 80 rods long contains 15 acres, while another field of the same width contains 9 acres; what is the length of the latter field? Ans. 48 rods.

NOTE.-In Ex. 9, the area is 6400 sq, rd., and each side is 80 rd.