MENSURATION OF SURFACES. Rectangle. 1. The side of a square floor is 24 ft. 6 in.; find its area. 2. Show how to find the area of a rectangle when the lengths of its sides are known. Example.-Find the acreage of a rectangular field which is a quarter of a mile long and 242 yards wide. 3. If the length of a rectangular field be 880 yards, and its breadth 440 yards, what is its area expressed in square miles? 4. Find the area in acres, etc., of a rectangle whose sides are 5 chains 14 links, and 6 chains 25 links. 5. How much land will be wanted for a road 8 miles long and 33 feet wide? 6. Find the side of a square whose area shall be 1 acre. 7. Find the area of a square whose side is 35 yards. 8. Find the area of a square whose side is 273 feet. 9. The sum of the perimeter and the diagonal of a square being 6 feet, find its side. 10. Find the side of a square whose area is equal to a rectangle 81 feet long and 60 inches wide. 11. The sides of a rectangle are 16 and 24 feet; what breadth of border must be taken off all round, that the remaining area may be 240 square feet? 12. Find the expense of carpeting a room 18 ft. 9 in. long and 17 ft. 6 in. broad, with carpet 27 inches wide, at 5s. 3d. per yard. 13. The cost of trenching a square garden whose side is 71 yards is £57, 15s. 24d.; how much is that per square yard? 14. The expense of paving a street half a mile long at 74d. per square yard was £430, 16s. 8d. Find the breadth of the street. 15. If a window be 8 feet 2 inches high by 5 feet 3 inches wide, how many panes of glass will fill it, if each pane measures 14 inches by 9 inches. 16. Find the cost of carpeting a room, 24 feet by 18 feet, with carpet 27 inches wide, at 5s. 4d. per yard. 17. A room is 36 feet long, 18 feet broad, and 12 feet high; find the whole cost of plastering the walls, at one shilling, and the ceiling at eighteen pence per square yard. 18. The rent of a square field at £2, 14s. 6d. per acre amounts to £27, 5s. od.; find the cost of putting a paling round the field, at 9d. per yard. 19. How many yards of paper, 27 inches wide, will be required to cover the 4 walls of a room 37 feet square, and 10 feet 6 inches high? 20. A court 20 yds. 2 ft. 6 in. long by 12 yds. 2 ft. 3 in. broad is paved with flagstones, each of which is 2 ft. 1 in. long, by 1 ft. 5 in. broad; how many are wanted? Explain your method as to a class. 21. How much paper, 21 inches wide, will be required to paper a room which is 24 feet long, 18 feet wide, and 12 feet high, allowance being made for a doorway which is 8 feet high by 4 feet wide, and for three windows, each of which is 6 feet high by 3 ft. wide. 22. A rectangular grass plot is 120 feet long by 80 feet wide, and it is surrounded by a gravel walk 6 feet wide; find the area of the gravel walk in square yards. 23. A man undertook to walk round a square field containing 13 acres 1089 yards in 7 minutes. If his pace was, on the average, 5 miles an hour, by how much time did he win? 24. The perimeter of a rectangular field, whose length is three times its breadth, is 800 yards; find its area and the distance from corner to corner. 25. Find the cost of making a road, a rod wide and 300 chains long, if the land costs £140 an acre, and the construction £4, 4. od. per square chain. Triangle (having given Base and Altitude). 1. Find the area of a triangle whose base is 10 and height 2. What is the height of an equilateral triangle whose sides are 10 ft.? Triangle (having given Three Sides). 1. Find (to three places of decimals) the area of the triangle whose sides are 15, 16, 17 inches. 2. What is the area of a triangle whose sides are 13, 15, and 14 feet respectively? 3. Find the area of a triangle whose sides are 848, 900, and 988 links. 4. How many acres are there in a triangular field whose sides are 600, 700, and 800 yards? 5. The three sides of a triangle are 500, 600, and 700 links; find its area in acres. 6. Find the area of a triangle whose sides are respectively 130, 140, and 150 yards long. 7. Find the area of a triangle whose sides are 3.27 ft., 4°36 ft., 5'45 ft. 8. What is the expense of reaping the corn in a field whose sides are respectively 150, 200, and 250 yards long, at 10s. an acre? 9. How many tiles a foot square will be required to pave a triangular court whose sides are 30, 40, and 50 ft. respectively? 10. A triangular field whose sides are 350, 440, and 750 yds. is let for £26, 5s.; what is that per acre? Rhombus. 1. Each side of a rhombus is 65 ft., and one of the diagonals is 104 ft.; find the area in square feet. 2. A four-sided field contains 40 acres, all the sides are equal, and one diagonal is twice as long as the other; find the length of a side in yards. Trapezoid. 1. Show how to find the area of a quadrilateral figure which has two sides parallel. Example. Find the area of a field having two parallel sides 650 and 925 links long, the perpendicular distance between the sides being 840 links. 2. If a field have two parallel sides, the sum of which is 1025 links, while the perpendicular distance between them is 310 links, what is the acreage of the field? 3. The parallel sides of a four-sided field are respectively 14 and 20 yards, and the perpendicular distance between them is 12 yards. What is its area? 4. The area of a trapezoid is 762*24 sq. ft., and the sum of the parallel sides is 158.8 feet; find the distance between them. 5. Two of the four sides of a field are parallel and differ by 4 ft. in length. Find them, their perpendicular distance being 19 ft., and the area of the field 475 square feet. 6. Find the area (in sq. chains) of a quadrilateral figure, of which two parallel opposite sides are respectively 750 links and 1225 links, and the perpendicular distance between these sides is 1540 links. Irregular Four-sided Figures. 1. In surveying a four-sided field ABCD, I measure AC = 190 and perpendiculars from the other corners upon AC = 45 ft. and 60 ft. respectively. Find its area. ft.; 2. ABCD is a quadrilateral field. AB measures 48 chains; BC, 20 chains; the diagonal AC, 52 chains; and the perpendicular from D on AC, 30 chains. Find the acreage of the field. 3. A four-sided field ABCD is measured by a chain from each corner to the corner opposite. AC is found to be 10 chains 11 links, and BD 11 chains 10 links; AC and BD are at right angles to one another. What does the field measure in a. r. p.? Circle, Semicircle, Sector, and Segment. 1. Find the area of a circle whose diameter is 48 yards. 2. Find the area of a circle of 72 feet radius. 3. The sides of a triangle are 13, 14, and 15 feet. Find the area of a circle which has the same perimeter. 4. If the perimeter of a semicircular flower bed be 60 feet, how many plants will it contain, allowing one square foot for each plant? 5. Find the diameter of a circular surface 6.68135 square feet in area. 6. Required the pressure (to the nearest hundredweight) on a circular plate 3 feet in diameter, the pressure on a square inch being 15 lbs. 7. What must be the diameter of a circular lawn in order that it may contain a quarter of an acre? 8. What is the diameter of a coach wheel that makes 600 revolutions in travelling a mile? 9. Give rules for finding the area of a circle and of the sector of a circle. 10. The area of a sector being 125 square feet, and of the whole circle 400 square feet, find the angle of the sector. 11. A segment of a circle subtends an angle of 60 at the centre, the radius being 12 feet; find the area of the segment. 12. The diameter of a circle is 12 feet; find the area of a square inscribed in it. 13. A circular court 80 yards in diameter has a circular grass plot, 64 yards in diameter, in the centre, with a road 8 yards wide all round; what will it cost to pave the road at 44d. the square foot? 14. The difference between the areas of two squares inscribed and circumscribed about a circle is 338 feet. Find the radius of the circle. Regular Polygons. 1. What is the length of the side of a heptagon which has an area of 1364 84 yards? 2. What is the length of the side of the octagon whose area is 30 1776 square feet? Irregular Polygons. 1. ABCDE is a five-sided field, and the angles at B, C, D are right angles; AB = 20 feet, BC= 18 feet, CD=32 feet, |