November 17th, 2014, 11:30 AM  #1 
Newbie Joined: Nov 2014 From: USA Posts: 4 Thanks: 0  Find Height of Spherical Cap
I need to find the height of a spherical cap, given the volume of the cap and the radius of the sphere. (Not the base radius.) Can anyone help me find the formula for that?

November 17th, 2014, 11:40 AM  #2 
Senior Member Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230  Then find it. Yes. What do you mean by "(Not the base radius)" ? Give an example. 
November 17th, 2014, 11:48 AM  #3 
Newbie Joined: Nov 2014 From: USA Posts: 4 Thanks: 0 
By that I mean I mean the radius of the initial sphere is given, and the volume of the spherical cap is given. I do not have the radius of the base of the spherical cap.

November 17th, 2014, 12:59 PM  #4 
Senior Member Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230  Did you mean something like this ? 
November 17th, 2014, 01:34 PM  #5 
Newbie Joined: Nov 2014 From: USA Posts: 4 Thanks: 0 
Close. It would be like my attached, with the volume and R given.

November 17th, 2014, 03:05 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,945 Thanks: 1136 Math Focus: Elementary mathematics and beyond 
The formula for the volume of a spherical cap in terms of the height of the cap and the radius of the sphere is $\displaystyle V=\frac{\pi h^2}{3}(3rh)$ Rearranging gives $\displaystyle h^33h^2r+\frac{3V}{\pi}=0$ which can be difficult to solve. You may wish to consider numerical methods for known values of $V$ and $r$. 
November 17th, 2014, 05:07 PM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,592 Thanks: 1038 
....easy (p = pi): solve p*h^33*p*r*h^2+3*v=0 for h  WolframAlpha 
November 18th, 2014, 02:32 AM  #8 
Newbie Joined: Nov 2014 From: USA Posts: 4 Thanks: 0 
Thank you.

November 18th, 2014, 05:15 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,592 Thanks: 1038 
$\displaystyle h^33h^2r+\frac{3V}{\pi}=0$ That hugelooking solution per Wolfram to solve the above equation can be made manageable this way (p = pi): a = SQRT[3V(3V  4pr^3)] b = 2pr^3  3V c = [(a + b) / (2p)]^(1/3) h = [r(r + c) + c^2] / c 

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