372. Cor. II. Let S and S' denote the areas of two whose radii are R and R', and diameters D and D', respect That is, the areas of two circles are to each other as the squares of their radii, or as the squares of their diameters. 373. Cor. III. The area of a sector is equal to one-half the product of its arc and radius. Given s and c the area and arc, respectively, of a sector of a whose area, circumference, and radius are S, C, and R, respectively. Proof. A sector is the same part of the O that its arc is of the circumference. 374. Cor. IV. Since similar sectors are like parts of the to which they belong (§ 369), it follows that Similar sectors are to each other as the squares of their radii. EXERCISES. 31. Find the circumference and area of a circle whose diameter is 5. 32. Find the radius and area of a circle whose circumference is 25 π. 33. Find the diameter and circumference of a circle whose area is 289 π. 34. The diameters of two circles are 64 and 88, respectively. What is the ratio of their areas? hext two PROP. XIV. PROBLEM. 375. Given p and P, the perimeters of a regular inscribed and of a regular circumscribed polygon of the same number of sides, to find p' and P', the perimeters of a regular inscribed and of a regular circumscribed polygon having double the number of sides. Solution. Let AB be a side of the polygon whose perimeter is p, and draw radius OF to middle point of arc AB. Also, draw radii OA and OB cutting the tangent to the O at Fat points A' and B', respectively; then, A'B' is a . side of the polygon whose perimeter is P. (§ 342) Draw chords AF and BF; also, draw AM and BN tangents to the O at A and B, meeting A'B' at M and N, respectively. Then AF and MN are sides of the polygons whose perimeters are p' and P', respectively. (§ 344) Hence, if n denotes the number of sides of the polygons whose perimeters are p and P, and therefore 2 n the number of sides of the polygons whose perimeters are p' and P', we have But OA' and OF are the radii of the polygons whose perimeters are P and p, respectively. 376. To compute an approximate value of (§ 367). (3) Solution. If the diameter of a O is 1, the side of an inscribed square is √2 (§ 352); hence, its perimeter is 2√2. Again, the side of a circumscribed square is equal to the diameter of the O; hence, its perimeter is 4. We then put in equation (2), Prop. XIV., We then put in equation (3), Prop. XIV., p = 2.82843, and P' = 3.31371. .. p' =√p × P' = 3.06147. These are the perimeters of the regular circumscribed and inscribed octagons, respectively. Repeating the operation with these values, we put in (2), P= 3.31371, and p= 3.06147. These are, respectively, the perimeters of the regular ciroumscribed and inscribed polygons of sixteen sides. In this way, we form the following table: The last result shows that the circumference of a O whose diameter is 1 is > 3.14157, and < 3.14163. Hence, an approximate value of # is 3.1416, correct to the fourth decimal place. NOTE. The value of π to fourteen decimal places is 3.14159265358979. to 30 EXERCISES. 35. The area of a circle is equal to four times the area of the circle described upon its radius as a diameter. 36. The area of one circle is 23 times the area of another. If the radius of the first is 15, what is the radius of the second ? 37. The radii of three circles are 3, 4, and 12, respectively. What is the radius of a circle equivalent to their sum? 38. Find the radius of a circle whose area is one-half the area of a circle whose radius is 9. 39. If the diameter of a circle is 48, what is the length of an arc of 85° ? 40. If the radius of a circle is 3 √3, what is the area of a sector whose central angle is 152° ? 41. If the radius of a circle is 4, what is the area of a segment whose arc is 120° ? (= 3.1416.) (Subtract from the area of the sector whose central is 120°, the area of the isosceles ▲ whose sides are radii and whose base is the chord of the segment.) 42. Find the area of the circle inscribed in a square whose area is 13.' 43. Find the area of the square inscribed in a circle whose area is 196 π. 44. If the apothem of a regular hexagon is 6, what is the area of its circumscribed circle ? 45. If the length of a quadrant is 1, what is the diameter of the circle? (π = 3.1416.) 46. The length of the arc subtended by a side of a regular inscribed dodecagon is π. What is the area of the circle ? 47. The perimeter of a regular hexagon circumscribed about a circle is 12 √3. What is the circumference of the circle? 48. The area of a regular hexagon inscribed in a circle is 24 √3. What is the area of the circle? 49. The side of an equilateral triangle is 6. Find the areas of its inscribed and circumscribed circles. 50. The side of a square is 8. Find the circumferences of its inscribed and circumscribed circles. 51. Find the area of a segment having for its chord a side of a regular inscribed hexagon, if the radius of the circle is 10. (=3.1416.) |
