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Math Help - overlapping area

  1. #1
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    overlapping area

    2 circles A and B with radius a and b units respectively ( suppose
    a < b ) touch at the point P . If circle A rotates x degree at P
    anti-clockwisely . ( where 0 < x < 180 )
    Find the area of the overlapping region of the 2 circles.
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  2. #2
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    Quote Originally Posted by yswong View Post
    2 circles A and B with radius a and b units respectively ( suppose
    a < b ) touch at the point P . If circle A rotates x degree at P
    anti-clockwisely . ( where 0 < x < 180 )
    Find the area of the overlapping region of the 2 circles.
    1. Draw a sketch.

    2. You'll get 2 sektors whose central angles are 2u and 2w respectively. You know that

    u+w=x

    The area of a segment is calculated by: Area of sector - area of right triangle.

    The common side of the two triangles is calculated by:

    a \cdot \sin(u) = b \cdot \sin(w)~\implies~\dfrac ab = \dfrac{\sin(w)}{\sin(u)}

    Unfortunately LaTeX doesn't work anymore. Therefore you wasn't able to see this line:

    \tan(w)=\dfrac{a \cdot \sin(x)}{b+a\cdot \cos(x)}

    Re-written into the simple syntax it means:

    tan(w)=(a * sin(x))/(b+a*cos(x))

    3. Calculating the orange area A_{orange}:

    A_{orange} = \underbrace{\dfrac{2u}{360^\circ} \cdot \pi a^2}_{sector} - 2 \cdot \underbrace{\dfrac12 \cdot a^2 \cdot \cos(u) \cdot \sin(u)}_{\text{area of right triangle}}

    4. Calculating the green area A_{green}:

    A_{green} = \dfrac{2w}{360^\circ} \cdot \pi b^2 - 2 \cdot \dfrac12 \cdot b^2 \cdot \cos(w) \cdot \sin(w)

    5. Add both areas and try to simplify this term a little bit.

    Maybe you can use the property: \sin(u) \cdot \cos(u)=\dfrac12 \left(\sin(2u) \right)
    Attached Thumbnails Attached Thumbnails overlapping  area-ueberlapp_sektoren.png  
    Last edited by earboth; April 18th 2011 at 10:29 PM.
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  3. #3
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    Thank you very much earboth! Your formula seems OK but can it be

    expressed in terms of a , b and x only?

    For special values of x :

    (1) If x = 0 , the area should be 0.

    (2) If x = 180 , the area should be a . a . pi sq.units. (i.e. area of circle A.)

    (3) If x = 90 , what will be the area ?

    Thanks again !
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  4. #4
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    Quote Originally Posted by yswong View Post
    Thank you very much earboth! Your formula seems OK but can it be

    expressed in terms of a , b and x only?

    For special values of x :

    (1) If x = 0 , the area should be 0.

    (2) If x = 180 , the area should be a . a . pi sq.units. (i.e. area of circle A.)

    (3) If x = 90 , what will be the area ?

    Thanks again !
    I've re-written those parts in my previous post which you didn't see because LaTeX doesn't work anymore. The line in red gives you the mathematical relation between a, b and x.
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  5. #5
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    Quote Originally Posted by earboth View Post
    I've re-written those parts in my previous post which you didn't see because LaTeX doesn't work anymore. The line in red gives you the mathematical relation between a, b and x.



    Hi earboth , after revising some basic rules of trigonometry ,
    I get the followings:
    Let c be the distance between the centres of the 2 circles,
    then c = √a*a+b*b-2ab cos(180-x)
    = √a*a+b*b + 2ab cos x
    Since a / c = sin w / sin (180 - x)
    = sin w / sin x
    Thus sin w = a * sin x / c
    i.e. w = arc sin ( a * sin x / c )
    Similarly u = arc sin (b * sin x / c )
    Substituting into your formula ,
    Orange area = u / 180 * π * a* a - a * a * sin (2u) / 2
    Green area = w / 180 * π * b* b - b * b * sin (2w) / 2

    Thus the total area can be expressed as a function of
    a , b and x only . ( But I wonder how you get the expression :
    tan(w)=(a * sin(x))/(b+a*cos(x))
    For a special value of x being 90 ,
    the overlapping area will be :
    arc sin (b * sin 90 / c )* π * a* a / 180
    + arc sin (a * sin 90 /c )* π * b* b / 180
    - the areas of the 2 triangles ( which = ab sq. units )
    = arc sin ( b / c ) * π * a* a / 180 + arc sin (a /c )* π * b* b / 180
    - ab
    = arc sin ( b / √a*a+b*b )* π * a* a / 180
    + arc sin ( a / √a*a+b*b )* π * b* b / 180
    - ab
    Am I correct ?
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  6. #6
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    Besides to find the overlapping area , we may also find the value

    of c ( i.e. the distance between the centres of the 2 circles) given

    by the formula : c = √a*a+b*b + 2ab cos x

    ( 1 ) If x = 0 , cos 0 = 1 , then c = √a*a+b*b + 2ab = a + b

    (2) If x = 180 ,cos 180 = - 1, then c = √a*a+b*b - 2ab = b - a

    If x is taken randomly from 0 to 180 , what will be the

    expected value of c ?
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