Sectors and Arcs

							Arc Length, Sectors, Sections
Geometry
Arc Lengths and Areas of Sectors
Arc Length

   The length of part of the circumference.
              The length of the arc depends on what two things?
                     1) The measure of the arc.
                       2) The size of the circle.




    An arc length measures distance while
    the measure of an arc is in degrees.
Sector of a circle

   A region bounded by 2 radii and an arc.




                    .
           Z                  Minor Arc
                           •Use 2 letters                    XZ
           9               •Angle is less than or equal to 180

    120°
            C                  Major Arc
                           •Use 3 letters                   XYZ
X                    Y     •Angle is greater than 180



    Central Angle:   Any angle whose vertex is the center of the circle



      m XZ = m<XCZ = 120o
     The measure of arc XZ equals the measure of angle XCZ
   Portions of a Circle: Determine the Arc measure based on the portion given.



                          180o                  120o
             90o
                                                                                     60o
       90o                180o                   120o                          60o



A.                   B.                   C.                       D.
¼ of a circle:       ½ of a circle:       1/3 of circumference :   6π out of a total 36π
   ¼ ● 360              ½ ● 360                                    on the circle:
                                               1/3 ● 360
                                                                        1/6 ● 360
   Area of a Sector Formula
measure of the central angle or arc




                                  m
            Area of a sector =
                                     πr 2
                                 360                 The area of the entire circle!


                       The fraction of the circle!

                                                                   .
   Arc Length Formula
measure of the central angle or arc




                                  m                  The circumference of the


                                     2πr
                                                                     entire circle!
               Arc Length =

                                 360
                       The fraction of the circle!

                                                                    .
                Find the length of AB and the area of sector AOB.

1.          A                  2.    240o                     3.   300o                      4. B            120o           5.                108o
                       90o                                                                                                        B

                90o                        240o                      300o 12                              120o                            108o
          O                B               O              A                              A            O                 A                               A
                 6                                 12                     O                                  2.4                         O 10√2

                                                                                     B
                                      B
     mAOB  90                mAOB  240                     mAOB  300                    mAOB  120                    mAOB  108
     radius  6                radius  12                     radius  12                    radius  2.4                   radius  10 2
 Fraction of circle:            Fraction of circle:            Fraction of circle:            Fraction of circle:                Fraction of circle:
                    ¼                               2/3                       5/6                                1/3                             3/10
Length of AB                   Length of AB                   Length of AB                   Length of AB                   Length of AB
Fraction ● circumference       Fraction ● circumference                                      Fraction ● circumference

     ¼ ● 12π                        2/3 ● 24π                      5/6 ● 24π                   1/3 ● 4.8π                   3/10 ● 20√2π
     3π units                       16π units                      20π units                   1.6π units                     6√2π units
Area of sector AOB             Area of sector AOB             Area of sector AOB             Area of sector AOB             Area of sector AOB
 Fraction ● area                    Fraction ● area            Fraction ● area                Fraction ● area                    Fraction ● area
     ¼ ● 36π                        2/3 ● 144π                     5/6 ● 144π                  1/3 ● 5.76π                   3/10 ● 200π
     9π units2                      96π units2                     120π units2                 1.92π units2                       60π units2
       28.26                              301.44                     376.8                            6.03                            188.4
6. The area of sector AOB is 48π and mAOB  270. Find the radius of ○O.

                           m
     Area of a sector =       πr2
                          360

                        270
                 48π =      πr2
                        360
                 4 16 3 2 4
                   48 =   r
                 3      4    3
                     64 =   r2
                      r=8
                             9
7. The area of sector AOB is    and mAOB  40 . Find the radius of ○O.
                             4

                            m
      Area of a sector =       πr2
                           360

                 9     40
                   π=      πr2
                 4     360
                 99 1 2 9
                     =   r
                 1 4 9      1
                   81 2
                     = r
                   4
                   r= 9
                       2
  Sections
Let’s talk pizza
        AREA OF SECTION =
AREA OF SECTOR – AREA OF TRIANGLE
     ¼ π r²    -   ½ bh
       Area of section =
area of sector – area of triangle
     ¼ π r²    - ½ bh

                     A OF    = ¼ 100π =
           10               25π
                     A OF    = ½∙10∙10=
                            50
A of circle = 100π    A OF SECTION =
                      25π - 50
           Find the area of the shaded region. Point O marks the center of the circle.

8.                          9.                    10.                    11.
                  60˚
                                                             12
              8                      O   6               O
                                                                               30
                                                                  60                    4
                                                                                     O




     160
           π units2              9π - 18 units2   24π - 36√3 units2      8π - 8√3 units2
      3
Some common fractions and measures!

Arc or Central Angle   Fraction of the Circle   Arc or Central Angle   Fraction of the Circle
      Measure                                         Measure


      36o                    1/10                    108o                     3/10
      60o                     1/6                     300o                    5/6
     120o                    1/3                      240o                    2/3

      30o                   1/12                      330o                 11/12
      45o                     1/8                     225o                    5/8