We then put in equation (3), Prop. XIV., p = 2.82843, and P' = 3.31371. .. p'√px P' = 3.06147. These are the perimeters of the regular circumscribed and inscribed octagons, respectively. Repeating the operation with these values, we put in (2), P=3.31371, and p = 3.06147. We then put in (3), p = 3.06147 and P' = 3.18260. .. p'=√p × P' = 3.12145. These are, respectively, the perimeters of the regular circumscribed and inscribed polygons of sixteen sides. In this way, we form the following table: The last result shows that the circumference of a O whose diameter is 1 is > 3.14157, and < 3.14163. Hence, an approximate value of π is 3.1416, correct to the fourth decimal place. NOTE. The value of π to fourteen decimal places is 3.14159265358979. EXERCISES. 35. The area of a circle is equal to four times the area of the circle described upon its radius as a diameter. 36. The area of one circle is 27 times the area of another. If the radius of the first is 15, what is the radius of the second? 37. The radii of three circles are 3, 4, and 12, respectively. What is the radius of a circle equivalent to their sum? 38. Find the radius of a circle whose area is one-half the area of a circle whose radius is 9. 39. If the diameter of a circle is 48, what is the length of an arc of 85° ? 40. If the radius of a circle is 3 √3, what is the area of a sector whose central angle is 152° ? 41. If the radius of a circle is 4, what is the area of a segment whose arc is 120° ? (3.1416.) (Subtract from the area of the sector whose central is 120°, the area of the isosceles A whose sides are radii and whose base is the chord of the segment.) 42. Find the area of the circle inscribed in a square whose area is 13. 43. Find the area of the square inscribed in a circle whose area is 196 π. 44. If the apothem of a regular hexagon is 6, what is the area of its circumscribed circle? 45. If the length of a quadrant is 1, what is the diameter of the circle? (3.1416.) 46. The length of the arc subtended by a side of a regular inscribed dodecagon is. What is the area of the circle? 47. The perimeter of a regular hexagon circumscribed about a circle is 12 v3. What is the circumference of the circle? 48. The area of a regular hexagon inscribed in a circle is 24 √3. What is the area of the circle? 49. The side of an equilateral triangle is 6. Find the areas of its inscribed and circumscribed circles. 50. The side of a square is 8. Find the circumferences of its inscribed and circumscribed circles. 51. Find the area of a segment having for its chord a side of a regular inscribed hexagon, if the radius of the circle is 10. (=3.1416.) 52. A circular grass-plot, 100 ft. in diameter, is surrounded by a walk 4 ft. wide. Find the area of the walk. 53. Two plots of ground, one a square and the other a circle, contain each 70686 sq. ft. How much longer is the perimeter of the square than the circumference of the circle? (= 3.1416.) 54. A wheel revolves 55 times in travelling diameter in inches? 1045 π ft. What is its If represents the radius, a the apothem, s the side, and k the area, prove that 55. In a regular octagon, s = r√2 s=r √2, a = } r√2 + √2, and k = 2 r2 √2. (§ 375) 56. In a regular dodecagon, s = r √ 2 − √3, a = ±r √2 + √3, and k = 3 r2. 57. In a regular octagon, S= 2 a (√2 − 1), r = a √ 4 − 2 √2, and k = 8 a2 ( √2 − 1). 59. In a regular decagon, a = r V10+2 √5. (§ 359.) (Find the apothem by § 273.) 60. What is the number of degrees in an arc whose length is equal to that of the radius of the circle? (= 3.1416.) (Represent the number of degrees by x.) 61. Find the side of a square equivalent to a circle whose diameter is 3. (T = 3.1416.) 62. Find the radius of a circle equivalent to a square whose side is 10. (3.1416.) 63. Given one side of a regular hexagon, to construct the hexagon. 64. Given one side of a regular pentagon, to construct the pentagon. (Draw a of any convenient radius, and construct a side of a regular inscribed pentagon.) 65. In a given square, to inscribe a regular octagon. (Divide the angular magnitude about the centre of the square into eight equal parts.) 66. In a given equilateral triangle to inscribe a regular hexagon. 67. In a given sector whose central angle is a right angle, to inscribe a square. Note. For additional exercises on Book V., see p. 231. APPENDIX TO PLANE GEOMETRY. MAXIMA AND MINIMA OF PLANE FIGURES. PROP. I. THEOREM. 377. Of all triangles formed with two given sides, that in which these sides are perpendicular is the maximum. Given, in A ABC and A'BC, AB = A'B, and ABL BC. Multiplying both members of (1) by 1 BC, BC × AB > BC × A'D. .. area ABC > area A'BC. (§ 312) 378. Def. Two figures are said to be isoperimetric when they have equal perimeters. PROP. II. THEOREM. 379. Of isoperimetric triangles having the same base, that which is isosceles is the maximum. Given ABC and A'BC isoperimetric A, having the same base BC, and ▲ ABC isosceles. To Prove area ABC area A'BC. Proof. Produce BA to D, making AD= AB, and draw line CD. Then, BCD is a rt. Z; for it can be inscribed in a semicircle, whose centre is A and radius AB. (§ 195) Draw lines AF and A'G 1 to CD; take point E on CD so that A'E = A'C, and draw line BE. Then since AABC and A'BC are isoperimetric, Now AF and A'G are the Is from the vertices to the bases ACD and A'CE, respectively. of isosceles ... CF = CD, and CG = 1⁄2 CE. .. CF CG. Multiplying both members of (1) by 1⁄2 BC, BC × CF > BC × CG. .. area ABC> area A'BC. (§ 94) (1) (?) |