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Math Help - Geodesic dome

  1. #1
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    Geodesic dome

    Maria and Fred decide to build a geodesic dome and to make life a little easier they make it spherical in shape, but only that portion of a sphere cut off by a plane, and less than a hemisphere.

    So now they need to put a roof on it. Maria measures the height h and the slice radius r and asks Fred to work out the surface area. "Thats impossible", says Fred. "Surely it will depend on R, the radius of the sphere and we don't know what that is."

    Who is right?
    ------------------------------
    Note: I know the answer. I just thought this would be a fun problem.
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  2. #2
    MHF Contributor
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    I'm going with BOTH. The sphere radius, R, can be calculated from h and r.

    R = h + \frac{r^{2}-h^{2}}{2h} = \frac{h^{2}+r^{2}}{2h}


    ...assuming I know what you mean by "Slice radius".
    Last edited by TKHunny; August 20th 2008 at 01:30 PM.
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  3. #3
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    Quote Originally Posted by TKHunny View Post
    I'm going with BOTH. The sphere radius, R, can be calculate from h and r.

    R = h + \frac{r^{2}-h^{2}}{2h} = \frac{h^{2}+r^{2}}{2h}

    ...assuming I know what you mean by "Slice radius".
    Yes, you are correct. Personally, though, I find myself siding with Maria, because, if all you know is the height of the dome h and the radius of its base, r, which I called the slice radius, the surface area of the curved part of the dome is just;

    \pi \, \left( {r}^{2}+{h}^{2} \right)

    and it seems both unnatural and unnecessary to figure out R, the radius of the sphere.
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  4. #4
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    You talked me into it. Fred said three things:

    1) Impossible
    2) Needs R
    3) Don't know R

    This is how I feel about them:

    1) Simply wrong.
    2) Correct, but maybe only indirectly.
    3) So? Calculate it.

    Maria it is! Go maria!!
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