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Perimeter/Area of Shaded Region within a Circle

  1. Dec 8, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the area and perimeter of shared region in the following diagram:
    upload_2016-12-8_23-38-46.png

    2. Relevant equations
    Area of shared Region =x/360 * PI * radius * radius
    Perimeter of shared Region = x/360 * 2* PI* radius



    3. The attempt at a solution
    I am finding the area of circle & then subtracting the area of triangle to find the area of shared region:
    Thus
    A of circle = Pi * radius * radius
    = 3.14 * 12 * 12
    = 144 PI
    A of Triangle (Note its an equilateral triangle) = sqr(s) * sqrt(3)/ 4 = 36 sqrt(3)
    Therefore ans = 144PI - 36 sqrt(3) but this is a wrong answer.

    Similarly perimeter of circle = 12 * 3 = 36
    Circumference of circle = 2 * PI * radius
    = 2* PI * 12
    =24 PI
    Therefore ans= 24PI - 36

    Some body please guide me why my logic is not coreect.

    Zulfi. upload_2016-12-8_23-38-46.png
     
  2. jcsd
  3. Dec 8, 2016 #2

    cnh1995

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    That will not give you the area of the shaded region. If you added the area of the triangle to the area of the shaded region, which area will you get?
    That's not how you calculate perimeter. You need to find minor arc length+CD.

    What is the formula for arc length which uses the given information in the diagram ?
     
  4. Dec 9, 2016 #3
    Hi,
    <That will not give you the area of the shaded region. If you added the area of the triangle to the area of the shaded region, which area will you get?>
    Thanks for your response.
    Is this not same as calculating area of Circle & then subtract the are of triangle from it. If not please guide me why?
    <What is the formula for arc length which uses the given information in the diagram ?>
    I gave all the formulae in the beginning:
    Area of shared Region =x/360 * PI * radius * radius
    Perimeter of shared Region = x/360 * 2* PI* radius


    Some body please guide me.
    Zulfi.
     
  5. Dec 9, 2016 #4
    path4136.png
    This ^_^.


    What is x ?
     
  6. Dec 9, 2016 #5

    Ray Vickson

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    Science Advisor
    Homework Helper

    There are formulas available for such areas; you just need to look them up on the internet, or look inside a textbook.

    Hint: look up "circular segment".
     
  7. Dec 9, 2016 #6
    He have enough information to find the area without formula.
     
    Last edited: Dec 9, 2016
  8. Dec 9, 2016 #7
    Hi,
    Thanks for your guidance. As you said perimeter is:
    <You need to find minor arc length+CD>.
    x/36 * 2* PI * radius + CD
    Note x= 60 which is the central angle & is equal to the angle of arc?? (Is this correct?)
    CD = 12 because its an equilateral triangle
    60/360 * 2 * PI * 12 + 12
    1/6* 2 PI * 12 + 12
    4*PI + 12

    Area of shaded region is:
    Area of Sector COD - Area of triangle COD
    60/360 * PI * RADIUS * RADIUS - 12 * 12 Sqrt(3)/4
    1/6 * PI * 12 * 12 -36 sqrt(3)
    24PI -36sqrt(3)
    Since the shaded region is part of sector O so it is not needed to calculate area of circle.
    Zulfi.
     
  9. Dec 9, 2016 #8

    cnh1995

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    Looks good.
     
  10. Dec 9, 2016 #9

    Ray Vickson

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    But he HAS been using formulas---just the wrong ones.
     
  11. Dec 9, 2016 #10
    Try finding the area of the slice of the circle and subtract the area of the triangle
    You have the angle so just try to find the right formulas to use
     
  12. Dec 13, 2016 #11

    epenguin

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    Gold Member

    Your calculation is far too unnecessarily complicated.
    For the triangle, forget about the circle. Just take a triangle like that by itself.
    Then for the <) shape, you have very probably seen processed cheeses (Philadelphia, Kraft are a couple of names to my knowledge) in a circular box. How many cheeses are there in one? :oldwink:
    (Or at least cheesees in a box or a slice of a pie -no pun intended - looking like your diagram.)
     
    Last edited: Dec 14, 2016
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