It is evident also that the length of the circle is greater than the perimeter of any inscribed polygon and that the length of the circle is less thay the perimeter of any circumscribed polygon. It is natural therefore to regard the successive perimeters of the regular inscribed polygons and also of the regular circumscribed polygons as better and better approximations to the length of the circle. (d) By careful computation it has been found that when the diameter of a circle is 1: Apparently when the diameter of a circle is 1, the length of the circle is approximately 3.1416. If we let C= the length of the circle and d = the length of the diameter, then C÷d= 3.1416. (e) By Proposition XI, § 386, the perimeters of regular polygons of the same number of sides have the same ratio as their radii, and hence as their diameters, and also as their apothems. If we double the diameter of the circle considered in part (d), then we shall obtain for the successive perimeters of the inscribed and of the circumscribed polygons exactly double the lengths given in part (d). Evidently then the length of a circle of diameter 2 is approximately double that of a circle of diameter 1; that is, C= 2 × 3.14166.2832. Again, C÷d= 3.1416. Similarly the length of a circle of diameter 5 is approximately 5 × 3.1416, or 15.7080. Again C÷d=3.1416. 389. The relation derived in parts (d) and (e) of § 388 is not only apparently true but can be proved to be true.. We shall assume it for the present. It amounts to assuming that the length of a circle bears to the length of its diameter a constant ratio. This fact is proved in § 415. The Greek letter π (pī) is used to denote this constant ratio. That is, C÷d=π, or С= πd. Two useful approximations of π are 3.1416 and 34. The length of a circle is called the Circumference of the circle. Note. The determination of the value of and of what sort of number is has been one of the most famous problems of mathematics. The Egyptians early recognized that C÷d is constant, and obtained for this ratio a value which corresponds to 3.1605. The Babylonians and Hebrews were content with the much less accurate value, 3. (See I Kings, vii. 23.) The method employed in this text was introduced by Antiphon (469– 399 B.C.), improved by Bryson (a contemporary, probably), and finally carried out arithmetically in a remarkable manner by Archimedes (287212 B.C.) in a pamphlet on the mensuration of the circle. Antiphon suggested the use of inscribed regular polygons of 4, 8, etc., sides as a means of approximating the length of the circle, and Bryson suggested using at the same time the corresponding circumscribed regular polygons. Archimedes employed inscribed and circumscribed regular polygons having 3, 6, ... 96 sides in his computation, and showed that T > 31 and <3. The methods employed by Archimedes remained for a long time the standard procedure in efforts to compute π. As mathematical skill increased, formulæ for T were derived, particularly in trigonometric form, which enabled diligent computers to obtain the value to more and more decimal places. Vieta (1540-1603) was the first to derive a formula for π (not, however, a trigonometric one). He gave for the value 3.141529653. Others carried out the computation to as many as 700 decimal places. A Holland mathematician, Huygens (1629-1695), at the age of twentyfive, proved some theorems which made it possible to improve greatly on the methods of Archimedes. He was able to obtain from a regular hexagon as accurate a value for π as Archimedes obtained from the regular 96-gon. Mathematicians were particularly interested in determining what kind of number π is. In 1766-1767, Lambert proved that it is not rational; i.e., that it cannot be expressed as the quotient of two integers. In 1882, through methods introduced by Hermite in 1873, Lindeman proved that π is a transcendental number; i.e., that it cannot be the root of an ordinary algebraic equation. This was the goal toward which previous efforts had been directed, and thus completely solved a problem to which many of the great mathematicians had given some attention. 390. Cor. 1. The circumference of a circle equals 2 πr, where r equals the number of linear units in the radius. 391. Cor. 2. The circumferences of two circles have the same ratio as their diameters or as their radii. Proof. Let ri, d1, and C1 be the radius, diameter, and circumference of one circle; and let r2, d2, and C2 be the radius, diameter, and circumference of another circle. Note. 11. = - Remember that this proof is based on an informal treatment. For the customary formal treatment of this theorem, read, if it seems desirable, § 414. Ex. 54. Find the circumference of a circle whose diameter is 5 in.; 8 in.; 10 in. Ex. 55. How long is the piece of rubber for the tire of a buggy wheel 4 feet in diameter ? Ex. 56. If the diameter of a circle is 48 in., what is the length of an arc of 85° ? Ex. 57. How long must the diameter of a circular table be in order to seat 20 people, allowing 30 in. to each person? (Express the result correct to the nearest inch.) Ex. 58. A fly wheel in an engine room has a diameter of 10 feet. Through how many feet does a point on its outer rim move in a minute if the wheel makes 100 revolutions per second? Ex. 59. (a) What is the diameter of a circular race track whose length along its inside edge is one mile? (b) If the track is 100 feet wide, determine the distance around it in the middle of the track. Ex. 60. Draw any circle. Construct the circle: (a) Whose circumference is 3 times that of the given circle. See § 391. (b) Whose circumference is that of the given circle. 392. Area of a Circle. In the adjoining circle are inscribed a square and a regular octagon. The area of the square is one half the product of its apothem and its perimeter. In symbols (§ 362) : H E B It is evident that the surface within each successive polygon is more nearly equal to the surface within the circle. On the other hand, it is clear that each successive apothem is more nearly equal to the radius and that the length of the polygon is more nearly equal to the length of the circle. (See § 388.) It is reasonable therefore to conclude that the area of a circle is one half the product of its radius and its circumference. Letting Krepresent the area of the circle, then K=1rx C. 393. Cor. 1. Since C = 2πr, then K = r × 2 πr = πr2. 395. Cor. 3. The areas of two circles have the same ratio as the squares of their radii or of their diameters. Letting K, and K, represent the areas of the circles whose diameters are d, and do, and whose radii are r1 and r2 respec tively, then This theorem was proved by Hippocrates (450-400 B.C.). Look Note. up his history. 396. A Sector of a Circle is the portion of the interior of a circle which is within a given central angle. The central angle is called the angle of the sector. 397. Cor. 4. The area of a sector is one half the product of the radius and the length of the arc intercepted by its angle. Sector A Let c the length of the arc and k = the area of the sector of a circle whose area, circumference, and radius are K, C, and r, respectively. Proof. The area of a sector has the same ratio to the area of the circle that the length of its arc has to the circumference; that is, X (1) 398. A Segment of a Circle is that portion of the interior of a circle which is between a chord of the circle and its subtended arc; as segment AXB, indicated by the shaded part of the adjoining figure. The area of a segment AXB may be determined by subtracting the area of ▲ AOB from the area of sector OAXB. Segment B Ex. 61. Find the circumference and area of a circle whose diameter is 5 in.; 8 in.; 10 in. Ex. 62. Find the radius and area of a circle whose circumference is 26 in.; 38 π in.; 15 in. Ex. 63. Find the radius and circumference of a circle whose area is 64 sq. in.; 81π sq. in.; 225 π sq. in.; 289π sq. in. Ex. 64. Find the side of a square equivalent to a circle whose diameter is 12 in. |