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How Does One Deduce Area of Intersection for Three Cirlces?

  1. Mar 26, 2015 #1
    1. The problem statement, all variables and given/known data
    upload_2015-3-26_16-41-20.png

    2. Relevant equations
    Most likely Acircle = πr2
    Not sure if there's others.
    3. The attempt at a solution
    I'm not sure where to start, I've never seen a question of this sort. They all have the same radius, hence same area, and each point/center is r away from another, but I don't know how, if at all, that gets me closer to understanding.

    Thanks for any help ahead of time!
     
  2. jcsd
  3. Mar 26, 2015 #2

    SteamKing

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    What can you say about triangle ABC?
     
  4. Mar 26, 2015 #3
    Well, I can see that triangle ABC must be equilateral, and so it must have three 60° angles. Is that what you're hinting at, or did I miss something? If yes, I'm still missing where to go next. The area of the triangle would be r2/2 and there would be the area of the small segments outside of the triangle left to find the entire area.
    Thanks!
     
  5. Mar 26, 2015 #4

    SteamKing

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    Is that your final answer for the area of triangle ABC?

    Maybe you should make a sketch before stating this area flat out.
     
  6. Mar 26, 2015 #5
    Sorry, that was an obvious error on my part, not sure what I was thinking. Area of the triangle should be 1/2 * r * √(r2 - r2/4) = r/2 * √[(4r2 - r2)/4] = r/2 * √(3r2/4) = r/2 * r/2 * √(3) = (r/2)2 * √(3) So, where do I go from here to integrate the π into my answer?

    Thank you very much for your help thus far!
     
  7. Mar 26, 2015 #6

    SteamKing

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    The triangle ABC covers most of the area of overlap amongst the three circles. Think about what it doesn't cover: namely the area outside the triangle ABC but inside the circular arcs AB, BC, and CA. How would you figure these areas?

    Hint: think about what the area is of a circular sector, for example one centered at point A and including arc BC. Since you know what the central angle of the arc BC is, what would be the area of this circular sector? Given that the area of the whole circle is πr2 and the central angle for the whole circle is 2π radians, what would be the area of a portion of the circle which had a central angle of θ radians?
     
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