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Homework Help: Intersecting circles and area

  1. Sep 17, 2010 #1
    1. The problem statement, all variables and given/known data

    I don't know why but my brain is having one of its moments and I can't work through this. Not even on paper anymore.

    Okay so I have 3 intersecting circles. Like a Venn Diagram. How do I find the area of all three minus the instersecting parts. I know how to do two intersecting, but I am trying to break this down into an algorithm to process it in my brain.

    This is not homework, but I am posting it here. I just need some help with sequencing the steps to solve it, and then my brain might click with again. I am mentally stuck with the application so need a kick start.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 20, 2010 #2
    You didn't specify whether the circles all had the same radius or not, so I'll assume they don't.

    By "3 intersecting circles. Like a Venn Diagram" I assume you mean that circle A partially intersects both circles B and C, circle B partially intersects both circles A and C, and circle C partially intersects both circles A and B so that there are 7 distinct areas bounded by the circles and their intersections (see attachment).

    Let's assume that circle A has a radius of r and it's center at (a,b)
    Also, circle B has radius s and center (c,d)
    And, circle C has radius t and center (e,f)

    The formulas for the 3 circles are then:
    Circle A: [itex]r^2 = (x-a)^2 + (y-b)^2[/tex]
    Circle B: [itex]s^2 = (x-c)^2 + (y-d)^2[/tex]
    Circle C: [itex]t^2 = (x-e)^2 + (y-f)^2[/tex]

    Let A be the area of circle A, B be the area of circle B, and C be the area of circle C.

    Call the intersection between circles A and B, area D (the football-like shape)
    Similarly, call the intersection between circles B and C, area E and the intersection between circles A and C, area F.

    Lastly, call the intersection of all 3 circles (the diamond-like shaped area in the center), area G.

    If I understand your question correctly, you are looking for A + B + C - D - E - F + 2G

    Note that when you subtract D, for instance, you are already subtracting area G at the same time. So, you again subtract area G when you subtract areas D and F. Therefore, you must add G back in twice.

    At this point, I'll leave the math to you.
     

    Attached Files:

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