214. The unit of measure for surfaces is a square each side of which is a unit of some known length. Thus, the unit of square inches is 1 square inch; of square feet, 1 square foot; of square yards, 1 square yard, etc. 215. The area of a rectangle is the surface included within the lines which bound it, and is expressed by the product of the two numbers representing the two dimensions, or by the number of times it contains a given unit of measure. The diagram represents a square yard, each side of which is 1 yd. or 3 ft. long, and the whole is divided into square feet, 1 sq. ft. being the unit of measure. In one row there are 3 sq. ft., in 3 rows there are 3 times 3 sq. ft., or 9 sq. ft. Hence the area of 1 sq. yd. is 9 sq. ft. 216. sq. ft. 8 sq. ft. x3=9 sq. ft. FORMULAS FOR RECTANGULAR SURFACES. 1. Length x breadth = area. length. sq. ft. The two given dimensions must be expressed in units of the same denomination. 217. Artificers, in estimating materials and labor, make use of the following units: The square foot for glazing, stone-cutting, flagging, and lumber. The square yard for plastering, paving, ceiling, and masonry. The square of 100 square feet, for roofing, flooring, and bricklaying. Shingles are estimated to average 16 in. long, and 4 in. wide, and are put up 4 bundles to the 1000. Allowing for waste, 1000 shingles, laid 4 in. to the weather, are estimated to cover a square. Laths are estimated at 4 ft. in length, and are put up 10 bundles to the 1000. 218. 1. What is the area of a floor 27 ft. by 18 ft. ? SOLUTION.-27 × 18 = 486; hence, the area is 486 sq. ft. (216, 1.) 2. What is the width of a hall that is 36 ft. long, and contains 252 square feet? SOLUTION,-256÷36 = 7; hence, the width is 7 ft. (216, 2.) 3. What is the length of a hall that is 8 ft. 4 in. wide, and contains 450 square feet? 4. How many tiles 8 in. square will be required to lay a floor 48 ft. by 10 ft. ? 5. What will be the cost of flagging a sidewalk 312 ft. long and 6 ft. wide, at $2.70 a square yard? 6. What will it cost to cement a cellar bottom 48 ft. 6 in. long and 27 ft. wide, at $.45 a square yard? 7. How many acres in a piece of land 60 rd. square? 8. What is the difference between the area of a floor 25 ft. square and that of two others, each 12 ft. 6 in. square? 9. What will be the cost of a piece of land 80 rd. long and 75 rd. wide, at $68 an acre? 10. A rectangular field 120 rd. long contains 24 A. What is its width? 11. Find the cost of covering the floor of a hall 45 ft. long and 6 ft. 6 in. wide with oil-cloth, at $1.35 a square yard? NOTE. In buying carpeting to carpet a room, it is necessary to decide whether the strips are to run lengthwise or across the room; also, how much must be turned under or cut off, at the side or end of the room, or both, on account of the dimensions of the room, or to properly match the figures. Multiplying the number of yards in each strip by the number of strips will give the whole number of yards required. 12. How many yards of carpeting 30 inches wide will carpet a floor 164 ft. long and 15 ft. wide, if the strips run lengthwise, and there be no loss from matching? 13. Find the cost of covering the floor of a hall 464 ft. long and 14 ft. 9 in. wide, with matting 14 yd. wide, at $.25 a yard. It will require 4 strips of matting, each 15 yd. long. 14. How many yards of velvet carpeting, 14 yd. wide, will cover a parlor floor 24 ft. 9 in. long and 17 ft. wide, if the strips run lengthwise, and the matching of the figures requires 9 in. to be turned under at one end of the room? 15. How many yards of Brussels carpeting, yd. wide, will cover the same floor, and what will be its cost, at $1.65 a yard, if the strips run lengthwise, and the matching of the figures. requires 6 in. to be turned under? 16. How many squares are there in a partition 104 ft. 9 in. long, and 20 ft. 4 in. high? 17. Find the cost of plastering the sides and ceiling of a room 40 ft. long, 36 ft. wide, and 221 ft. high, at $.36 a sq. yd., allowing 1375 sq. ft. for doors, windows, and baseboard. 18. How many planks 16 ft. long and 9 in. wide, will be required to floor a room 36 ft. long and 24 ft. wide? 19. Find the cost of glazing 6 windows, each 8 ft. 3 in. by 5 ft. 4 in., at $.75 a square foot. 20. How many sods, each 16 in. square, will be required to turf a yard 53 ft. 4 in. long and 28 ft. wide? 21. How many shingles averaging 4 in. wide, laid 6 in. to the weather, will cover the roof of a building 46 ft. long, each of the two sides being 20 ft. wide from the eaves to the ridge, the first course on each side being double? 22. What will be the cost of wainscoting a room 21 ft. 8 in. by 14 ft. 10 in., and 10 ft. 6 in. high, at $.30 a sq. yd.? 23. Find the cost of slating a roof 64 ft. 9 in. long and 45 ft. wide, at $15.37 per square? 24. What is the area of a square field, the distance around which is 560 rods? 25. Find the cost of painting a roof 52 ft. long and 30 ft. wide, at $.75 a square. SOLIDS. 219. A solid has three dimensions, length, breadth, and thickness or height. 220. A rectangular solid is a body bounded by six rectangu lar faces. Rectangular Solid. Cube. A cube is a rectangular body whose faces are all equal. 221. The unit of measure for solids is a cube, the edge of which is a unit of some known length. Thus, the unit of cubic inches is a cube, the edge of which is 1 inch, or 1 cubic inch; of cubic feet, 1 cubic foot, etc. 222. The volume or solid contents of a rectangular body is the space included within the surfaces which bound it, and is expressed by the product of the numbers representing its three dimensions, or by the number of times it contains a given unit of measure. which are 27 cu. ft. Hence the volume of 1 cu. yd. is 27 cu. ft. So the volume of a solid, formed of two adjacent sections, is expressed by 3 cu. ft. x 3x2 = = 18 cu. ft. 223. FORMULAS FOR RECTANGULAR SOLIDS. 1. Length x breadth x height = volume. 2. Volume 3. Volume (length x height) = breadth. 4. Volume (breadth x height) = length. The three given dimensions must be expressed in units of the same denomination. For measures of volume, the following units are used: The cubic foot for bricklaying, masonry, and hewn timber. The cubic yard for embankments, excavations, and masonry. The perch of stone, 16 ft. long, 1 ft. wide, and 1 ft. high, equal to 24 cu. ft. It is customary, however, to call 25 cu. ft. a perch. Brick.-The size of a common brick is 8×4 × 2 in., and for ordinary calculation it is sufficiently accurate to reckon 27 bricks to the cubic foot, laid dry, or 22 laid in mortar. Brickwork is generally estimated by the thousand bricks. In estimating material, allowance is made for openings in walls, as doors, windows, etc. In estimating labor, the length of each wall is measured on the outside, and thus each corner is measured twice. Sometimes, by special contract, an allowance is made for one half the openings and corners. |