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Math Help - Area of a segment in a circle

  1. #1
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    Area of a segment in a circle

    I was wandering if anybody could help with this, it would be fully appreciated:

    Two parallel lines are drawn 2cm from the centre of a circle of radius 4cm. Calculate the area between them.

    I know to find the area of the circle first[16pi], but where do I go next?

    Once again, thanks in advance,
    D Loizou
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  2. #2
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    Nobody?
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  3. #3
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    Hello, loizoud94!

    Two parallel chords are 2cm from the centre of a circle of radius 4cm.
    Calculate the area between them.
    Code:
                  * * *
              *:::::|:::::*
            *:::::::|:::::::*
          A*- - - - + - - - -*B
              *           *
          *      * 120°*      *
          *         *O        *
          *         |         *
                    |
          C*- - - - + - - - -*D
            *:::::::|:::::::*
              *:::::|:::::*
                  * * *

    The area of the circle is: . \pi(4^2) \:=\:16\pi cm².

    The area between the chords is:
    . . the area of the circle minus the area of the two shaded segments.


    We find that the sector has a central angle of 120°.
    Hence, the area of the sector is: . \tfrac{1}{3}\pi r^2 \:=\:\tfrac{16}{3}\pi

    The area of \Delta AOB is: . \tfrac{1}{2}(4)(4)\sin120^o \:=\:4\sqrt{3}

    . . Hence, the area of a segment is: . \tfrac{16}{3}\pi - 4\sqrt{3}


    The area betwen the chords is: . A \;=\;16\pi - 2\left(\tfrac{16}{3}\pi - 4\sqrt{3}\right)

    Therefore: . A \;=\;\tfrac{16}{3}\pi + 8\sqrt{3} cm²

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