Plane and solid geometry . 558. Def. The area of a circle is the common limit w^iichthe successive areas of inscribed and circumscribed regularpolygons approach as the number of sides is successively in-creased and ettch side approaches zero as a limit. BOOK V 267 Proposition XIV. Theorem 559. Th e area of a circle is equal to one half the prod-uct of its circumference and its Given circle 0, with radius R, circumference C, and area K. To prove K=\ CR, 1. Argument Circumscribe about circle O a regularpolygon. Call its perimeter P andits area S. Then S = ^PR. As the number of side
Plane and solid geometry . 558. Def. The area of a circle is the common limit w^iichthe successive areas of inscribed and circumscribed regularpolygons approach as the number of sides is successively in-creased and ettch side approaches zero as a limit. BOOK V 267 Proposition XIV. Theorem 559. Th e area of a circle is equal to one half the prod-uct of its circumference and its Given circle 0, with radius R, circumference C, and area K. To prove K=\ CR, 1. Argument Circumscribe about circle O a regularpolygon. Call its perimeter P andits area S. Then S = ^PR. As the number of sides of the regularcircumscribed polygon is repeatedlydoubled, P approaches C as a limit. .*. i PR approaches i Ci? as a limit. 5. Also S approaches Z as a limit. 6. But S is always equal to ^ PR. 7. .*. ir = i CR. 4. Reasons§ 517, b. §557.§550. §561. § 2. 7. § 355. 560. 37te product of a variable and a constant is a variable. 561. TJie limit of the product of a variable and a constant, notzero, is the limit of the variable multiplied by the constant. (Proofs of these theorems will be found in the Appendix, §§ 585 and590.) 562. Cor. I. The area of a circle is equal to vlf. Hint. K= iC-B = i ??ZttB • R = irB^. 268 PLANE GEOMETRY 563. Cor. II. The areas of two circles are to ea/^h otheras the squares of their radii, or as the squares of tlxeir
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