326. COR. 1.- The circumferences of circles are to each other as their diameters. or By (166) the above proportion becomes c:C::2r:2R, c:C::d: D. 327. COR. 2.- The ratio of the circumference of a circle to its diameter is a constant quantity. This constant ratio is usually denoted by ", the Greek letter p, called pi. The numerical value of can be found only approximately, as can be proved by the higher mathematics. Hence, in any circle, the circumference and its diameter are incommensurable. 328. COR. 3.- The circumference of a circle equals the diameter multiplied by π. THEOREM V. 329. The area of a regular polygon equals half the product of its perimeter and apothem. Let P be the perimeter and A the apothem of the regular polygon MNORQS. To prove that the area of MNORQS = } P × A. Draw CO, CR, CQ, etc., dividing the polygon into as many As as it has sides. All the As have the common altitude A, and the sum of their bases equals P; THEOREM VI. 330. The area of a circle equals half the product of its circumference and radius. Let C be the circumference and R the radius of the O O. G To prove that the area of the circle = CXR. Inscribe a regular polygon, and denote its perimeter by P, and its apothem by A. Then the area of the polygon = } P × А. (329) Now, this is true whatever may be the number of sides of the polygon; hence it is true when the number is infinitely great, in which case P = C, and A == R; ... the area of the O = C × R. E. D. 331. COR. 1.- The area of a ○ = π R2, R being the radius. 332. COR. 2.-The area of a sector of a circle equals half the product of its arc and the radius. DEFINITION. 333. Similar Sectors are sectors of different circles, which have equal angles at the centre. THEOREM VII. 334. Circles are to each other as the squares of their radii. Letrand R denote the radii of the Oso and 0. 335. COR.-Similar sectors are to each other as the squares of their radii. PROBLEMS IN CONSTRUCTION. PROBLEM I. 336. To inscribe a square in a given circle. Let O be the centre of the given O. Draw any two diameters, as AB and CD, ⊥ to each other, and draw AC, CB, BD, and AD. Now the angles about the centre are equal; ... the circumference is divided into four equal arcs; ... the chords AC, CB, BD, and AD are equal. The Ls ADB, DBC, BCA, and CAD are Ls; (188) (208) ... E. F. ADBC is the required square. (123) 337. COR. To inscribe a regular polygon of 8 sides, bisect the arcs subtended by the sides of an inscribed square and draw chords; and by continuing the process, we can inscribe regular polygons of 16, 32, etc., sides. |