Surface of a spherical cap

  1. bobie

    bobie 682
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    from wiki:
    Suppose we have a hemisphere of radius 10 r10 (a) and cut it in ten horizontal slices (1 is on the top), does that mean that all slices have the same surface ?

    even slice 1 has surface 62.8 (2\pi *10*1)? and its a (r1) =4.36?
    so, the area of slice 4 (like all others) is
    2pi*10*4-2pi*10*3 = 2pi*10= 62.8

    is this correct?
    If it is not, what is the formula to find the area and a (r1) of slice 1?

    Thanks
     
    Last edited: Jul 24, 2014
  2. jcsd
  3. Simon Bridge

    Simon Bridge 15,261
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    It is possible to cut the horizontal slices so that each has the same surface area by varying the spacing.
    To work out the surface area of each slice - use calculus.
     
  4. bobie

    bobie 682
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    Hi Simon, thanks. I am just starting to learn calculus.
    If I understood what you said, if we cut 10 equal slices of 1 cm , they will not have the same surface?
    Could you show me how to frame the equation(s)?
    Thanks
     
  5. Simon Bridge

    Simon Bridge 15,261
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    That's right - I would be surprised if the areas came out the same.

    If we say that the floor is the x-y plane and up is the +z axis, then you start by dividing the whole hemisphere (radius R) into very thin disks - thickness "dz". Then you want to work out the equation for the surface area "dS" of the disk between z and z+dz in terms of z and R.
     
  6. bobie

    bobie 682
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    Then I misinterpreted wiki?

    Because if we find the area of the slice on the 'floor' S 9 subtracting the cap with h = 9 (2pi*10*9) = 565.48 from the hemisphere 628,3 we get 62.8
    and the same happens all the way to the top to S1
    Wher did I go wrong?
     
  7. Simon Bridge

    Simon Bridge 15,261
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    Hah - I just tried it out and I am surprised ;) - try of for 2 slices.
    I still think your best proof involves doing the calculus.
     
    1 person likes this.
  8. bobie

    bobie 682
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    I was, too, that's why I checked here, it seems amazing, right!
    If you are intrigued, check by calculus, and let me know!
     
  9. 1 person likes this.
  10. Simon Bridge

    Simon Bridge 15,261
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    Yes. This is one of the reasons I like to answer questions here - sometimes someone surprises me.
    This is the sort of thing that is obvious in retrospect.
     
  11. bobie

    bobie 682
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