DE=13 feet. Find the length of the remaining side and area of the field. 2. In a five-sided yard ABCDE, I note the following measurements: AB=14, BC=7, CD=10, DE=12, EA = 5, AC 17, all in chains; also the angle at E is a right angle. Find the area. Miscellaneous Exercises. 1. Write out the table of square measure, and calculate how many square yards there are in a farm which covers I square mile, 30 acres, and 6 square chains. 2. Write out a table of square measure, and express a square chain in square yards, and an acre in chains and square links. square 3. Write down the rules in the following cases :— (1) The circumference of a circle being given, to find the diameter. (2) To find the area of a trapezoid. (3) To find the area of a segment which is less than a semicircle. 4. The distance between two places on a map drawn on the scale of of an inch to a mile is 5 feet 2 inches. Find the real distance. 5. A barn is 60 feet long, and 30 feet wide, and 20 feet high measured perpendicularly from the level of the eaves. How many square feet of slating does it contain. 6. Give the rule-(1) for finding the area of a triangle when the three sides are known; (2) for finding the diameter of a circle when the height of an arc and the chord of half the arc are known; (3) Simpson's rule for finding the area of a figure bounded by straight lines and a curve. 7. The area of an equilateral triangle is 100 square feet; find its perimeter in yards. 8. The sides of a triangle are 51 feet, 68 feet, and 85 feet; show that it is right-angled, and that the ratio of the areas of the triangles into which it is divided by a perpendicula from the right angle on the hypotenuse is 9: 16. 9. A yard whose area is 810 square feet is to be paved with 1080 rectangular tiles; find the breadth (which is three-fourths of the length) of each tile. 10. The sides containing the right angle of a triangle are 9 feet and 12 feet; prove that the areas of the equilateral triangles described on them are together equal to the area of the equilateral triangle described on the hypotenuse. 11. What is the length of the sides of a square field, whose area is equal to that of a triangular field, with sides of 182, 168, and 70 yards respectively? 12. Find the side of a square inscribed in a semicircle whose radius is 5 feet. 13. The shape of a glass mirror is a semicircle on a square whose side is 5 feet; find its cost at 62 shillings per square foot. 14. The sides of a right-angled triangle are 70 feet and 98 feet; find, in square feet, the area of the semicircle described on the hypotenuse (π=37). 15. A window in the form of a square with a semicircular head contains 414 square feet; what is its extreme height? 16. The perpendicular from the right angle of a right-angled triangle on the hypotenuse divides the triangle into two parts, one of which is double of the other; show that the perpendicular trisects the hypotenuse. 17. A room 25 feet 3 inches long, 14 feet 6 inches wide, has a semicircular bow 21 feet in diameter thrown out on one side; find the area of the whole room. 18. A person intended to buy a Brussels carpet of a yard wide, price 4s. 6d. per yard, for a room 40 feet by 28; but afterwards decided to put down a square Turkey carpet, each side of which is 20 feet, in the middle, and a drugget a yard wide in the remainder of the room. The Turkey carpet costing £25, find the price of the drugget per yard so that the whole expense may be the same as for the Brussels carpet. 19. Find the ratio of the area of a circle to the area of a regular hexagon which has the same perimeter. 20. Find the area of a regular hexagon inscribed in a circle whose radius is 50 feet. 21. In the width of a circular court 30 feet in diameter is a well in the form of a regular hexagon, each side of which is 2 feet. Find the expense of paving the court at 2s. 3d. per square foot. 22. Give a method of finding the area of any rectilineal figure; and find the perpendicular distance between the parallel sides of a trapezoid whose area is 3 acres, the sum of the parallel sides being 11 chains. 23. The length of an arc of a circle being given, how do you find the number of degrees in the angle subtended by the arc at the centre? 24. A circle is described about a square whose side is 6 yards; what part of a square chain is the area between the square and the circle? Circle, Semicircle, Sector, and Segment. 1. 1,809 5616 sq. yds. 2. 16,286 0544 sq. ft. 3. 140 26205 sq. ft. 4. 213. 6. 136 323 cwts. 7. 39 215 yds. 1. 19:37 yds. 8.2 ft. 9. See solutions. 10. 112° 30'. 11. 13:07 sq. ft. 12. 72 sq. ft. 13. £305, 9s. 8‡d. 14. 13 ft. Regular Polygons. | 2. 25 ft. Irregular Polygons. 1. 13 ft. 546 sq. ft. 2. 1425567 acres. |