9. The axis of a parabola is 30, and the base 24; shew that the area found by seven equidistant ordinates from the axis differs from the true area by 3.2857; but if calculated by seven equidistant ordinates from the base, we get the area exactly. IO. The diameter of a semicircle is 30, and an ordinate divides it into two parts, 12 and 18. Calculate the areas of the two parts into which the semicircle is divided; and shew that their sum differs from the true area by 3.1063. First ordinates: 0, 74833, 10'1980, 12, 13.2665, 141421, 146969. Second ordinates: 0, 9, 12, 13 7477, 146969, 15, 146969. First area, 1309730; second area, 219°4920; sum, 350 4650; area of semicircle, 353'5714; difference, 3'1063. REGULAR FIGURES. XXI. To find the area of a regular figure. Multiply the square of the side by the corresponding number in the following table :— Ex. Find the area of a regular decagon whose side is 7 ft. 3 in. 7'25 7:25 3625 1450 5075 52.5625 7.694 2102500 4730625 3153750 3679375 404'4158750 EXAMPLES. I. Find the area of a regular pentagon whose side is 263 yards. 2. Find the cost of boarding an heptagonal floor whose side is 18 feet, at is. 6d. per square yard. Ans. £9. 16s. 23d. Find the area of a regular duodecagon whose side is 7 ft. 9 in. MISCELLANEOUS EXAMPLES. I. Find the value of 34. 7. 8 square feet of carpet at 35. 7 d. per foot. Ans. £6. 5s. 612d. 2. Find the side of a square the area of which is equal to a rectangle whose sides are 21 ft. 11 in. and 197 ft. 3 in. Ans. 65 ft. 9 in. 3. Find the hypothenuse of a a right-angled triangle whose base is 4 ft. 9 in. and perpendicular 14 ft. 8 in. Ans. 15 ft. 5 in. 4. Find the area of a square board whose side is 27 ft. 8 in., and the cost of the same at 7s. 83d. per square Area, 85.0.7.1.4 yds. Cost, £32. 17s. 41d. yard. 5. is 24 ft. Find the area of a right-angled triangle whose base 8 in. and perpendicular 36 ft. 3 in. Ans. 447 ft. 6. Find the area of a rectangular wall whose length is 113 ft. 9 in. and height 33 ft. 6 in., and the cost of whitewashing the same at 1d. per yard. Area, 423.4. 10 yds. Cost, £2. 125. 113d. 7. The hypothenuse of a right-angled triangle is 46 ft. 5 in., and the base 13 ft. 9 in.; find the perpendicular. Ans. 44 ft. 4 in. 8. The statute acre contains 4840 square yards, and the lineal perch 5 yards; shew that the statute acre contains 160 poles. 9. Find the area of a triangle whose base is 12 ft. 8 in. and perpendicular 8 ft. 8 in. Ans. 54. 10.8 ft. IO. is 25. II. Find the area of an equilateral triangle whose side Find the area of a triangular field whose sides are 20, 30, 40 chains. 12. The area of a triangle is 19 a. 3 r. 8 p., and the perpendicular is 18 chains; find the base. 13. Ans. 22 chains. The area of a trapezium is 14 a. 2 r. 16 p., and the sum of the perpendiculars upon the diagonal from the -opposite angles is 1600 links; find the diagonal. Ans. 1825 links. 14. The hypothenuse of a right-angled triangle is 114 ft. 8 in. and the base 96 ft. 7 in.; find the perpendicular. Ans. 61.9.8 feet. 15. Find the cost of boarding a floor in the form of a trapezoid whose parallel sides are 16 ft. 8 in., and 14 ft. Io in., and the distance between them 8 ft. 4 in., at 3d. per foot. Ans. £1. 18s. 3 d. 16. The hypothenuse of a right-angled triangle is 37 ft. 6 in., and the other two sides are in the ratio of 3 to 4; find the sides. Ans. 22 ft. 6 in., 30 ft. 0 in. 17. Shew that the square pole contains 625 square links. 18. How many dozens of paper 21 inches wide will be required for a room 24 ft. 9 in. long and 9 ft. 4 in. high? Ans. 33 19. Find the cost of carpeting a floor 18 ft. 6 in. long, and 15 ft. 4 in. wide, with carpet for the centre at 3s. 6d. per yard; having a border 3 feet wide; at 2s. 6d. per yard. Ans. £4. 135. 9d. 20. Find the circumference of a circle whose diameter is 6 ft. 3 in. 3 in. 21. 22. Find the area of a circle whose diameter is 19 ft. The hypothenuse of a right-angled triangle is 163 ft. 9 in., and the other sides are in the ratio of 2 to 3; find the Ans. 90 ft. 10 in., 136 ft. 3 in. sides. 23. Find the area of a triangle whose sides are 12, 15, 17 chains. 24. The area of a triangle is 30 perpendicular is 6 ft. 4 in.; find the base. Ans. 8 a. 3 r. 4 p. square feet, and the 25. Find the circumference of a circle whose diameter is 18 inches. Ans. 4 ft. 8 in. 26. The radius of a circle is 36 inches; find the length of an arc which contains 40°. Ans. 2 ft. 1 in. 4 in. 27. Find the side of a square whose diagonal is 15 ft. 28. Find the area of a trapezoid whose parallel sides are 24 ft. 9 in. and 32 ft. 7 in., and the distance between them 16 ft. 4 in. Ans. 469.2.8 ft. 29. 30. Find the cost of the following boards: II ft. 2 in. × I ft. 9 in.,,, 2d. 37 ft. 0 in. x 2 ft. 1 in., 21d. 90 ft. 8 in. x 1 ft. 7 in.,,, 23d. Prove that 10 square chains make an acre. 31. The diagonal of a four-sided field is 50 chains, and the perpendiculars upon it from the opposite angles are 11 chns. 15 lks. and 15 chns. 18 lks.; find the area. Ans. 68 a. 3 r. 32. A ladder 50 feet long stands upright against a wall. How far will the top fall by drawing out the bottom 14 feet? Ans. 2 feet. |