# Height of a spherical cap with known volume?

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1. Feb 26, 2015

### streeters

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. Feb 26, 2015

### SteamKing

Staff Emeritus
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.

3. Feb 26, 2015

### Ray Vickson

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 .