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Math Help - why is the area of the intersection of the cylinder and the plane Pi?

  1. #1
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    why is the area of the intersection of the cylinder and the plane Pi?

    Question
    I have a cylinder equation (x-1+az)^2+(y+bz)^2=1. The intersection of
    the cylinder and the YOZ plane should be bigger than the base when it
    is an ellipse. The base is the circle (x-1)^2+y^2=1 with area Pi. The
    intersection is (az-1)^2+(y+bz)^2=1. I tried different a's and b's,
    The area is always Pi, for example letting a=1 and b=10. The
    calculation is done by using Wolfram|Alpha: Computational Knowledge Engine. The short axis of
    this ellipse seems smaller than 2. I think since the
    base circle has radius 1, the ellipse should have short axis 2.

    The result is very counter intuitive. When one cuts the cylinder
    with a plane and the result is an ellipse, the area of the ellipse
    should be bigger than the base circle.
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  2. #2
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    Re: why is the area of the intersection of the cylinder and the plane Pi?

    I'd like to remove the second paragraph. It is not true for an oblique cylinder. I still want to know if there are some a and b to make the intersection bigger than Pi.
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  3. #3
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    Re: why is the area of the intersection of the cylinder and the plane Pi?

    I found it if the wolfram website is right. When a=.1 and b=.2, the area is about 10Pi.
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