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Math Help - Area between Circles

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    Area between Circles

    What is the area between two circles, radius one, that go through each other's centres?

    Not sure if this requires calculus or not. I couldn't work it out. *
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    Quote Originally Posted by Aquafina View Post
    What is the area between two circles, radius one, that go through each other's centres?

    Not sure if this requires calculus or not. I couldn't work it out. *
    One quarter of the area in question is the difference between a sector (radius = 1, central angle = 60) and a right triangle. (see attachment)
    Attached Thumbnails Attached Thumbnails Area between Circles-zweieckflaeche.png  
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    Quote Originally Posted by earboth View Post
    One quarter of the area in question is the difference between a sector (radius = 1, central angle = 60) and a right triangle. (see attachment)
    Don't you mean the Area of the triangle + the sector?
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    referring to the attached pic:
    both triangles ABC and DBC can be proved equilateral and congruent to each other.
    the area of sector ABC
    = (1/6)(pi)(1^2)
    = pi/6
    and the area of triangle ABC
    = (1/2)(AB)(BC)(sin60)
    = (1/2)(1)(1)(3^0.5 / 2)
    = 3^0.5 / 4
    the difference between these two areas is the area of each one red region (in attachment), let A be it.

    hence, the total area of overlapped region of the two circles
    = area of triangle ABC x 2 + 4A
    = 3^0.5 /4 x 2 + 4(pi/6 - 3^0.5 / 4)
    = 1.2284 sq. units
    Attached Thumbnails Attached Thumbnails Area between Circles-temp.bmp  
    Last edited by ukorov; November 2nd 2009 at 06:17 PM.
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    Quote Originally Posted by Aquafina View Post
    Don't you mean the Area of the triangle + the sector?<<<<< No
    1. The area of the sector (That's the area with the blue borderline) is

    a_{sector} = \dfrac16 \cdot \pi \cdot 1^2 = \dfrac16 \cdot \pi

    2. The area of the right triangle (That's the lightgrey area) is

    a_{triangle} = \dfrac12 \cdot \dfrac12 \cdot \dfrac12 \cdot \sqrt{3} = \dfrac18 \cdot \sqrt{3}

    3. The white area is

    a_{quart} = a_{sector} - a_{triangle} = \dfrac16 \pi - \dfrac18 \sqrt{3}

    4. The area in question is

    a = 4 \cdot a_{quart} = \dfrac23 \pi - \dfrac12 \sqrt{3} \approx 1.22837
    Attached Thumbnails Attached Thumbnails Area between Circles-zweieckflaeche.png  
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