6. Required the length of the arc, whose chord is 6, the Ans. 6'11706. radius being 9. PROBLEM XI. To find the area of a circle. The area of a circle may be found from the diameter and circumference together, or from either of them alone, by the following rules. RULE I.* Multiply half the circumference by half the diameter. Or, Take of the product of the whole circumference and diameter. RULE 2.† Multiply the square of the diameter by '7854. * DEMONSTRATION. A circle may be considered as a regular polygon of an infinite number of sides, the circumference being equal to the perimeter, and the radius to the perpendicular. But the area of a regular polygon is equal to half the perimeter multiplied by the perpendicular, and consequently the area of a circle is equal to half the circumference multiplied by the radius, or half the diameter. Q. E. D. + DEMONSTRATION. All circles are to each other as the squares of their diameters. (Euc. XII. 2.) But the area of a circle, whose diameter is (by Rule 1.) Therefore 12: d2 :: '7854, &c. : 1, is 7854, &c. '7854, &c. x d2 12 ='7854, &c. Xd area of a circle, whose diameter is d. Q. E. D. The following propositions are those of Metius and Archimedes. GGg RULE 3. Multiply the square of the circumference by '07958. RULE 4. As 14 is to 11, so is the square of the diameter to the area. RULE 5. As 88 is to 7, so is the square of the circumference to the area. As 452: 355 :: square of the diameter: area. If the circumference be given, instead of the diameter, the area may be found as follows. The square of the circumference x'07958 area. area. As And if d be the diameter, c the circumference, a the area, and 3'14159, &c. then The following table will also show most of the usual problems, relating to the circle and its equal or inscribed square. 1. Diameter x '8862 side of an equal square. = 2. Circumference 2821 side of an equal square. = 3. Diameter x 7071-side of the inscribed square. EXAMPLES. 1. To find the area of a circle, whose diameter is 10, and circumference 31'4159265. 4. Circumference x 2251 side of the inscribed square. 5. Area X 6366= side of the inscribed square. 6. Side of a square × 1'4142= diameter of its circumscribed circle. 7. Side of a square × 4'443 circumference of its circumscribed circle. 8. Side of a square × 1'128= diameter of an equal circle. 9. Side of a square × 3'545= circumference of an qual circle, 2. Required the area of the circle, whose diameter is 7, and circumference 22. Ans. 381 3. What is the area of a circle, whose diameter is 1, and circumference 3'1416? Ans. 7854. 4. What is the area of a circle, whose diameter is 7? Ans. 38°4846. 5. How many square yards are in a circle, whose diameter is 3 feet? Ans. 1'069. 6. How many square feet does a circle contain, the cir cumference being 10'9956 yards? J Ans. 86°19266 PROBLEM XII. To find the area of a sector of a circle. RULE I.* Multiply the radius, or half the diameter, by half the arc of the sector for the area. Or, take of the product of the diameter and arc of the sector. NOTE. The arc may be found by Problem X. RULE 2.t As 360 is to the degrees in the arc of the sector, so is the whole area of the circle to the area of the sector. NOTE. For a semicircle, take one half; for a quadrant, one quarter, &c. of the whole circle. EXAMPLES, 1. What is the area of the sector CAB, the radius being 10, and the chord AB 16? * The rule for finding the area of the sector is evidently the same, as that for finding the area of the whole circle. † DEMONSTRATION. Let r radius, d➡ number of degrees in the arc of the sector, and A= its area. Then will 4r2×'7854=r2 × 31416 area of the whole cir cle, and 2r X3'1416-its circumference. |