2 circular measure   arc length
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2 circular measure arc length

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This presentation shows how to find the arc length and area of sector of a circle

This presentation shows how to find the arc length and area of sector of a circle

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2 circular measure   arc length 2 circular measure arc length Presentation Transcript

  • Circular Measure
    Arc Length and Area of a Sector
  • Review: Radian Measure
    In general, if the length of arc, s units and the radius is r units, then
    That is the size of the angle (θ) is given by
    the ratio of the arc length to the length of the radius.
    For example:
    If s = 3 cm and r = 2 cm, then
  • Arc Length
    From our definition of the radian, we have:
    s = rθ
    where θ is in radians
    For example:
    If θ = 2.1 radians and r = 3 cm
    Length of arc AB, s = rθ
    = 3 × 2.1 cm
    = 6.3 cm
  • Example 1
    The diagram shows part of a circle, centre O, radius r cm. Calculate:
    The value of r,
    BOC in radians.
    In the sector AOB, s = 2.4 cm and θ= 1.2 radians
    In the sector BOC, s = 1.4 cm and r= 2 cm
  • Area of a Sector
    In the diagram, the angle of the sector AOB is θradians.
    By proportion:
    r
    A
    s
    Let the area of sector AOB be A.
    Thus,
    r
    Now, as s = rθ, we have:
  • Example 2
    In the diagram, arc Ab and CD are arc of concentric circles, centre ). If OA = 6 cm, AC = 3 cm and the area of sector AOB is 12 cm2, calculate
    AOB in radians,
    The area and perimeter of the shaded region.
    Let be AOB = θ radians
    Area of the sector AOB = 12 cm2
    (b) Now OC = OA + AC = 9 cm
    Area of the shaded region
    = area of sector COD – area of sector AOB
    = 27 – 12 = 15 cm2
  • Arc Length and Area of a Sector
    where θ is in radians
  • Connection: Area of Triangle
    where C is an acute angle
  • Area of Triangle
    where C is an obtuse angle
  • Example 3
  • Solution:
  • Problem
  • Solutions