Plane and solid geometry . ven circle 0 of unit diameter, AB half the side of a reg-ular circumscribed polygon of n sides, and CB half the sideof a regular circumscribed polygon of 2 n sides; denote AB by I and CB by |. To find X in terms of s. Argument 1. Dra\r CO and AO. ^ 2. /. BOC=\Z. BOA. 3. .-. in A OAB, AC : CB = AO : BO, 4. But AC = AB — CB, 5. And A o-v 6. .-. AB — CB : CB AB + BO- = ^AB^-{-~b7/: bo. 7. Substituting - for AB, - for CB, and - 2 2 2 AV2y 8. « — X : ic = V?^l : 1. Reasons 1. § 54, 15. 2. § 517, h, 3. §432. 4. § 54, 11. 5. §446. 6. §309. 7. §309. 8. §403. BOOK V Argument
Plane and solid geometry . ven circle 0 of unit diameter, AB half the side of a reg-ular circumscribed polygon of n sides, and CB half the sideof a regular circumscribed polygon of 2 n sides; denote AB by I and CB by |. To find X in terms of s. Argument 1. Dra\r CO and AO. ^ 2. /. BOC=\Z. BOA. 3. .-. in A OAB, AC : CB = AO : BO, 4. But AC = AB — CB, 5. And A o-v 6. .-. AB — CB : CB AB + BO- = ^AB^-{-~b7/: bo. 7. Substituting - for AB, - for CB, and - 2 2 2 AV2y 8. « — X : ic = V?^l : 1. Reasons 1. § 54, 15. 2. § 517, h, 3. §432. 4. § 54, 11. 5. §446. 6. §309. 7. §309. 8. §403. BOOK V Argument F. Reasons 9. 10. ., s — X = X Vs- -f 1. s 9. § Solving: 1 + Vs^ + 1 271 Ex. 1017. Given a circle of unit diameter and an inscribed and acircumscribed square ; compute the side of the regular inscribed and the regular circumscribed octagon. Proposition XVII. Problem 568. To compute the approximate value of the circum-ference of a circle in terms of its diameter; to computethe value of Given circle ABCD, with unit diameter. To compute approximately the circumference of circle ABCDin terms of its diameter; to compute the value of tt. Argument 1. The ratio of the circumference of a circle to its diameter is the same forall circles. 2. Since the diameter of the given circle is unity, the side of an inscribed squarewill be \ V2. Reasons 1. §553. 2. §522. 272 PLANE GEOMETRY Argument 3. By using the formula x = ^^ ~ ^\~ ^ the sides of regular inscribed polygonsof 8, 16, 32, etc., sides may be com-puted ; and by multiplying the lengthof one side by the number of sides,the length of the perimeter of eachpolygon may be obtained. The re-sults are given in the table below. 4. Likewise if the diameter of the given circle is unity, the side of a circum-scribed square will be 1. 5. By using the formula x = 1 + Vs^+i the sides of regular circumscribedpolygons of 8,16, 32, etc., sides maybe computed; and by multiplying thelength of one side by the
Size: 1586px × 1576px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
