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Height of a spherical cap with known volume?

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  1. Feb 26, 2015 #1
    1. The problem statement, all variables and given/known data

    What equation is needed to calculate the height of a spherical cap with a fixed volume and radius?

    2. Relevant equations

    V=πh/6 (3a^2 + h^2)
    Where V = volume, h is cap height, a is cap radius

    3. The attempt at a solution

    I have tried to separate the h out and got as far as:
    6V/π = h[(√3a+h)^2 -2√3ah]

    but think I am on the wrong track
     
  2. jcsd
  3. Feb 26, 2015 #2

    SteamKing

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    An alternate formula for the volume of a spherical cap can be found here:

    http://en.wikipedia.org/wiki/Spherical_cap

    V = (πh2/3)*(3r - h), where h is the height of the cap and r is the radius of the sphere.
     
  4. Feb 26, 2015 #3

    Ray Vickson

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    Your equation for ##h## is a cubic. There are formulas for solutions of cubic equations, so you can get ##h## in terms of ##a## and ##V##. In fact, you have a so-called "depressed cubic" where ##h## appears only in ##h^3## and ##h## terms (with no ##h^2## term), and for such equations the solution is a bit simpler. See, eg., http://en.wikipedia.org/wiki/Cubic_function or
    http://www.sosmath.com/algebra/factor/fac11/fac11.html .
     
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