Question

Finding the Height of a Spherical Cap with it's Base Radius and Volume?

Hi,

I am trying to figure out how to find the height of a spherical cap while only knowing it's base radius and it's volume. I have found a formula to find it height using the sphere's initial radius (the sphere that the spherical cap has been laterally cut from) but I can not figure out how to find the radius without the height and nor can I find the height without the radius. Any help would be greatly appreciated.

Parker

Answers

Wikipedia has the following formula:

V = πh(3a2+ h2) / 6

... for the volume of the cap given the base radius a and cap height h. Solving this for h requires solving a cubic equation:

h^3 + 3ah - 6V/π = 0

I don't see an easy way to factor this.

WolframAlpha gives this gory solution for the only real root:

x = (sqrt(pi^2 a^3+9 V^2)+3 V)^(1/3)/pi^(1/3)-(pi^(1/3) a)/(sqrt(pi^2 a^3+9 V^2)+3 V)^(1/3)

I had to rename the unknown from h to x to get Alpha to recognize it as a cubic to solve. Anyway, if Wikipedia is right, and WolframAlpha is accurate, any other approach will give an answer that is equivalent to this. So, either there's no easy answer, or this is half of a truly impressive identity.

#1

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#2