MEASURES OF SURFACE. 304. Two lines meeting at a point form an angle. Thus the lines ab and b c form the angle a b c. 305. The lines are the sides of the angle, and the point where they meet is the vertex. b An Angle. 306. The size of an angle is the amount by which one side is turned away from the other. Thus the angle def is greater than the angle a bc, for the side ef is turned away from ed more than b c is turned away from ba. 307. When one line meets another so Thus, e C An Angle. d as to form two equal angles, each of these angles is a right angle, and the lines are perpendicular to each other. the line b c is turned away equally from ba and bd, making the angles a b c and cbd equal to each d other. d b Right Angles. 308. A flat surface, as the surface of a slate or the top of a table, is a plane surface. 309. A rectangle is a plane surface bounded by four straight lines, and having all its angles right angles. A Rectangle. 310. A square is a rectangle all of whose sides are equal. 311. A square each of whose sides is 1 inch long is a square inch; a square each of whose sides is 1 foot long is a square foot, etc. 312. The area of a surface is its contents reckoned in square units. A Square. To find the Area of a Rectangle. 313. ILLUSTRATIVE EXAMPLE. If the length of a rectangle is 4 inches and its breadth is 3 inches, how many square inches does it contain ? ILLUSTRATION. Explanation. A rectangle which is 4 in. long and 1 in. wide will contain 4 square inches, and a rectangle of the same length and 3 in. wide must contain 3 times 4, or 12 square inches. (See illustration.) In the same way it can be shown that the area of any rectangle is found by multiplying the number of units in the length by the number of like units in the breadth. This is expressed, for brevity, as multiplying the length by the breadth. 314. Oral Exercises. a. How many square inches are there in a rectangle 8 in. long and 5 in. wide? 11 in. long and 10 in. wide? 12 in. (1 foot) long and 12 in. (1 foot) wide? Then how many square inches are there in a square foot? b. How many square feet are there in a rectangle 8 ft. long and 7 ft. wide? How many square feet are there in a square whose sides are each 3 ft. (1 yard) long? c. How would you find the number of square yards in a square whose sides are each 5 yards or 1 rod long? 315. In measuring surface we employ the square mile (sq. m.), acre (A.), square rod (sq. rd.), square yard (sq. yd.), square foot (sq. ft.), and square inch (sq. in.). These are the units of MEASURES OF VOLUME. 316. A rectangular solid is a solid bounded by six rectangles. 317. The rectangles are the faces of the solid, and, together, make its surface. The bounding lines of the solid are its edges. A Rectangular Solid. 318. A cube is a rectangular solid bounded by six equal squares. A cube each of whose edges is 1 inch long is a cubic inch. A cube each of whose edges is 1 foot long is a cubic foot, etc. 319. The volume of a solid is its contents reckoned in cubic units. A Cube. To find the Volume of a Rectangular Solid. 320. ILLUSTRATIVE EXAMPLE. What is the volume of a block of marble 4 ft. long, 2 ft. wide, and 3 ft. thick? Explanation. If the block is 4 feet long and 2 feet wide, its lower base must contain 4 X 2, or 8 square feet (Art. 313). A solid 1 foot thick upon these square feet will contain 8 cubic feet, and a solid 3 feet thick will contain 3 times 8 or 24 cubic feet. In the same way it can be shown that the volume of any rectangular solid is found by multiplying the number of units ILLUSTRATION. in the length by the number of like units in the breadth, and this product by the number of like units in the thickness. This is expressed, for brevity, as multiplying together the length, breadth, and thickness. 321. Oral Exercises. a. How would you find the number of cubic inches in a cube 12 inches long, 12 inches wide, and 12 inches thick, or in 1 cubic foot? b. How many cubic feet in a cube 3 feet long, 3 feet wide, and 3 feet thick, or in 1 cubic yard? 322. In measuring solids we employ the cubic yard (cu. yd.), cubic foot (cu. ft.), and cubic inch (cu. in.). These are the units of Wood is generally cut for the market into sticks 4 feet long, and laid in piles, so that the length of the sticks becomes the width of the pile. A pile 4 feet wide, 4 feet high, and 8 feet long, contains 1 cord. One eighth of a cord is called 1 cord foot. 1 cord foot contains 16 cubic feet. (See illustration above.) c. How many cords are there in a pile of wood 4 feet wide, 4 feet high, and 20 feet long? 32 feet long? 90 feet long? d. What is the cost of a pile of wood 2 feet wide, 4 feet high, and 10 feet long, at $8 a cord? e. What must I pay for 3 cords of hard wood at $9.50 a cord, and of a cord of pine at $5 a cord? MEASURES OF WEIGHT. 323. In weighing grocer- 324. In weighing silver, ies and most other common goods, we use the ton (T.), pound (lb.), and ounce (oz.). These are the units of Avoirdupois Weight. 16 oz. 1 lb. 2000 lb. 1 T. NOTE I. In weighing some articles, as iron and coal at the mines, and goods on which duties are paid at the United States custom-houses, the long ton of 2240 lbs. is used. In this weight 28 lb. = 1 quarter (qr.). 4 qr. = 1 hundredweight (cwt). 20 cwt. 1 T. gold, precious stones, etc., we use the pound, ounce, pennyweight (pwt.), and grain (gr.). These are the units of Troy Weight. 24 gr. = 1 pwt. 20 pwt. = 1 oz. 12 oz. = 1 lb. 325. Comparison of Weights. 7000 gr. 66 = 1 lb. av. Which is heavier, a pound Troy or a pound avoirdupois? an ounce Troy or an ounce avoirdupois? NOTE II. For apothecaries' weight, see Appendix, page 307. NOTE III. The standard unit of weight is the TROY POUND. From this the other units of weight are derived. NOTE IV. A cubic foot of water weighs 62 lbs., or 1000 oz. avoirdupois. 326. Oral Exercises. a. How many ounces in 1 lb. avoirdupois? in 2 lb. 1 oz. ? b. How many ounces in 1 lb. Troy? in 4 lb. 5 oz.? c. Change 50 gr. to pennyweights; 90 pwt. to ounces. d. How many ounces in 3 pounds of silver? e. What is the value of a gold chain weighing 11⁄2 ounces at 90% a pwt.? f. At 80 a pound for camphor, what is the cost of an. ounce ? g. How many more pounds in a long ton than in a common ton? |