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Math Help - Find the area

  1. #1
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    Question Find the area

    Find the area of the crescent-shaped region(called a lune) bounded by arcs of circles with radii r and R(see the figure)
    Find the area-untitled.png
    thank you very much
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  2. #2
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    There is not enough information. You also need to know the distance between centers of the two circles.
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  3. #3
    MHF Contributor chisigma's Avatar
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    If R and r are known the distance d between the centers is...

    \displaystyle d=R\ \cos \sin^{-1} \frac{r}{R} (1)

    Kind regards

    \chi \sigma
    Last edited by chisigma; November 14th 2010 at 08:07 AM.
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  4. #4
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    Oh, yes! I missed the little "right angle" symbol on the r radius!
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  5. #5
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    The area of the lune is the area of the upper half of the small circle, plus the area of the triangle formed by the two radii of the large circle and the diameter of the small circle (see the attachment), minus the area of the sector of the large circle bounded by those two radii. So the area of the lune is

    \boxed{\tfrac12\pi r^2 + r\sqrt{R^2-r^2} - R^2\arcsin(r/R)}.

    (Seeing this problem in the Calculus section, I started trying to calculate the area as a double integral using polar coordinates. It's possible to do it that way, but the geometric approach is far easier!)
    Attached Thumbnails Attached Thumbnails Find the area-lune.png  
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