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Volume of two pieces of a sphere cut by a plane

  1. May 9, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the unit sphere [itex]x^{2} + y^{2} + z^{2} = 1[/itex]

    Find the volume of the two pieces of the sphere when the sphere is cut by a plane at [itex]z=a[/itex].

    3. The attempt at a solution

    My interpretation is that [itex]a[/itex] is a point on the z-axis that the plane cuts at. So the height of the segment is [itex]r-(r-a)[/itex]. After that, I'm not sure how to proceed. Should you somehow integrate the volume of the segment between [itex]r=1[/itex] and [itex]r=a[/itex]?
    Last edited: May 9, 2013
  2. jcsd
  3. May 9, 2013 #2
    Volume or area? You say volume in the title and area in the question.
  4. May 9, 2013 #3
    Volume. Edited accordingly.
  5. May 9, 2013 #4
    Well, you will have ##a<1## (since it's a unit sphere) so you will want to go the other way around; but, yes, what you are proposing should work. If you give a try and get stuck, show your work in lots of detail and we'll be able to help more.
  6. May 9, 2013 #5


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    The height of the segment is r-a. Your expression of the height, r - (r - a), equates to just a.
  7. May 9, 2013 #6


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    Obviously a must be between -1 and 1. It is sufficient to assume that [itex]0\le a\le 1[/itex] and calculate the volume above z= a and below the sphere. (Of course, the volume of the entire sphere is [itex](4/3)\pi[/itex] so the volume low z= a is [itex](4/3)\pi[/itex] minus the volume above. And if a< 0, just flip it over.)

    In cylindrical coordinates the base is given by [itex]r^2+ a^2= 1[/itex] so the cover that base r goes from 0 to [itex]1- a^2[/itex] and [itex]\theta[/itex] from 0 to [itex]2\pi[/itex]. For each r and [itex]\theta[/itex], the height is [itex]z- a= \sqrt{1- r^2}- a[/itex]
  8. May 14, 2013 #7
    Ok I believe I have that one figured out. The next problem now is to calculate the surface area of those segments using spherical coordinates. I'm told the formula [itex]S = \int^{b}_{\phi=a}\int^{d}_{\theta=c} sin\theta d\theta d\phi[/itex] should be used. What are a, b, c and d here?
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