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Math Help - elliptical footprint of a round beam

  1. #1
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    elliptical footprint of a round beam

    Hi,

    I have some problems with a basic geometric problem (it is a long time ago I had some mathematics)
    How do I calculate the surface area of a elliptical footprint of a beam that falls into the surface under a 45 degree angle? I only have the diameter D of the incident beam.
    I got:
    sqrt(2) pi (D/2)^2

    is this correct?

    thanks,
    Ilindema
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  2. #2
    MHF Contributor

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    Quote Originally Posted by ilindema View Post
    Hi,

    I have some problems with a basic geometric problem (it is a long time ago I had some mathematics)
    How do I calculate the surface area of a elliptical footprint of a beam that falls into the surface under a 45 degree angle? I only have the diameter D of the incident beam.
    I got:
    sqrt(2) pi (D/2)^2

    is this correct?

    thanks,
    Ilindema
    Start by drawing an isosceles right triangle The hypotenuse is the image of the beam and one of the legs is a diameter of the beam. That is, c^2= D^2+ D^2= 2D^2 so c, the major axis of the ellipse, is D\sqrt{2}. The minor axis is, of course, the diameter of the beam, D.

    The area of an ellipse with major and minor semi-axes a and b is \pi ab so, yes, the area of this ellipse is \pi \left(\frac{D}{2}\right)\left(\frac{D}{2}\sqrt{2}\  right), exactly what you have.
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  3. #3
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    thanks a lot! Now I have some peace
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