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Circles and arcs

Chapter 10.6. Circles and arcs. Circle. A set of all points equidistant from the center. Center. Circle. A circle is named by the center. P. Circle P ( P). Diameter. A segment that contains the center of a circle and has both endpoints on the circle. Diameter. Radius.

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Circles and arcs

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  1. Chapter 10.6 Circles and arcs

  2. Circle • A set of all points equidistant from the center Center

  3. Circle • A circle is named by the center P Circle P (P)

  4. Diameter • A segment that contains the center of a circle and has both endpoints on the circle. Diameter

  5. Radius • A segment that has one endpoint at the center of the circle and the other on the circle. Radius

  6. Congruent Circles • Congruent circles have the congruent radii P Q

  7. Central Angle • An angle whose vertex is the center of the circle. Central Angle

  8. Arc • Part of a circle. From point to point on the outside of the circle. Arc

  9. Semicircle • An arc that’s half of the circle. 1800 Semicircle • Has a measure of 1800

  10. Minor Arc • A minor arc is smaller than half the circle. 400 Minor Arc • Same measure as the corresponding interior angle

  11. Major Arc • A major arc is larger than half the circle. 3200 400 Major Arc • 360 minus the minor arc

  12. Practice 1 Name 3 of the following in A. 1. the minor arcs 2. the major arcs 3. the semicircles

  13. Adjacent Arcs • Adjacent arcs are arcs of the same circle that have exactly one point in common.

  14. Arc Addition Postulate • The measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs. 700 1100 400

  15. Practice 2 • Find the measure of each arc in R. • UT • UV • VUT • ST • VS

  16. Practice 3 • Find each indicated measure for D. 1. mEDI 2. 3. mIDH 4.

  17. Circumference • The distance around the circle • A measure of length

  18. Circumference • The circumference of a circle is π times the diameter (a = πd) or 2 times π and the radius (a = 2πr). Diameter

  19. Circumference C = d • Example: = 4 or = 12.52 D = 4

  20. Circumference C = 2r • Example: = 2(5) = 10 r = 5 or = 31.4

  21. Practice 4 • Find the circumference of each circle. Leave your answer in terms of . 1. 2.

  22. Arc Length • The length of an arc is calculated using the equation: 600 measure of the arc ________________ circumference * 360

  23. Arc Length • The length of an arc is calculated using the equation: 600 measure of the arc ________________ d * 360

  24. Arc Length • The length of an arc is calculated using the equation: 600 measure of the arc ________________ 2r * 360

  25. Arc Length 600 7 measure of the arc ________________ d * 360

  26. Arc Length 600 7 60 ________________ 7 * 360

  27. Arc Length = 3.67 600 7 1 ________________ 22 * 6

  28. Practice 5 • Find the length of each darkened arc. Leave your answer in terms of . 1. 2.

  29. Area of a Circle • The product of π and the square of the radius. Radius A = r2

  30. Area of a Circle A = r2 • Example: = 52 = 25 r = 5 or = 78.54

  31. Practice 6 • Find the area of a circle: • 6 in. radius • 10 cm. radius • 12 ft. diameter

  32. Sector of a Circle • A sector of a circle is a region bounded by an arc of the circle and the two radii to the arc’s endpoints. • You name a sector using the two endpoints with the center of the circle in the middle.

  33. Sector of a Circle • Sector is the area of part of the circle Area of blue section

  34. Area of Sector of a Circle • The area of a sector is: measure of the arc ________________ r2 * 360

  35. Sector of a Circle • Find the area of the sector 600 12

  36. Arc Length 600 12 measure of the arc ________________ r2 * 360

  37. Arc Length 600 12 60 ________________ 122 * 360

  38. Arc Length = 24 600 12 1 ________________ 144 * 6

  39. Segment of a Circle • Part of a circle bounded by an arc and the segment joining its endpoints

  40. Area of a Segment of a Circle • Equal to the area of the sector minus the area of a triangle who both use the center and the two endpoints of the segment.

  41. Area of a Segment of a Circle • Sector – Triangle = Segment - =

  42. Area of a Segment of a Circle • Find the area of the segment. 600 12

  43. Area of a Segment of a Circle • Separate the triangle and the sector 600 600 12 12

  44. Area of a Segment of a Circle • Find the area of both figures 600 600 12 12

  45. Area of Sector = 24 600 12 60 ________________ 122 * 360

  46. Area of Triangle Find the base 6ð3 or 10.4 6 600 Find the altitude 12

  47. Area of Triangle a = ½bh 10.4 6 600 12 = ½(12)(10.4) = 62.4

  48. Area of a Segment of a Circle • Subtract the triangle from the Sector 24 62.4 - 13 = 24 62.4

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