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I found few equations here:

Developing one ($V_{\text{cap}}=\frac{1}{3}\pi h^{2}(3R-h)$) and considering that $R$ in my case is equals to $1$ and $V_{\text{cap}}$ is $0.4$ I can't figure out $h$.

I am stuck here: $0.4 = h^2 (3-h)$

Tried also to use various equation solvers but they all return 0.

Thanks a lot!

P.S.: I hope this doesn't sound as silly as it feels.

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Substituing $R=1$ and $V_{\text{cap}}=0.4$ in the formula $$V_{\text{cap}}=\frac{1}{3}\pi h^{2}(3R-h)$$ gives $$0.4=\frac{1}{3}\pi h^{2}(3-h).$$ –  Américo Tavares Mar 3 '12 at 11:03
... from which you get the cubic equation $$5\pi h^{3}-15\pi h^{2}+6=0.$$ –  Américo Tavares Mar 3 '12 at 11:08

1 Answer 1

up vote 1 down vote accepted

If I interpreted your question correctly then you simply do as Américo Tavares does and substitute in the values. Given that $V_{cap}=0.4$ and $R=1$ you get: $$V_{cap}=\frac{1}{3}\pi h^2(3R-h)\Longrightarrow0.4=\frac{1}{3}\pi h^2(3-h)$$ Simplifying gives: $$0=5\pi h^3-15h^2+6$$ However, unfortunately it seems as though the zero's of the cubic seem to have rather nasty closed forms which can be found here.

An alternate method of finding the volume of a cap can be used using a little bit of calculus.

Say we have a circle with radius $r$ centered at $r$. Then we get a circle with equation $(x-r)^2+y^2=r^2$.

Here is an example with radius 3 with center $(3,0)$ enter image description here

We can simplify the general equation and get $y^2=r^2-(x^2-2xr+r^2)=2xr-x^2$. Since the volume of revolution of a circle is a sphere, the function for its volume is $$\int\pi y^2dx=\int \pi(2xr-x^2)dx$$ Now, if you want to find the volume of a cap with height $x$ you get: $$V_{cap}(x)=\int_0^x\pi(2tr-t^2)dt=\pi r x^2-\frac{\pi x^3}{3}$$

Edit: My "alternate method" is in fact the exact same method as that givien on the link you provided, just with a slightly different proof. I lieu of this, please consider only the first part of the answer.

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Hi Emile thanks a lot for the explanation. So, given the radius of the sphere as 1 and the volume of the cap 0.41888, how can I practically determine x (h)? –  Nuthinking Mar 3 '12 at 12:25
If you are seeking for practicality a graphic calculator, or if you don't have one, wolfram alpha will do. In any case, the solution you should get needs to be less than the diameter and has to be positive. That excludes (in the case of the link I provided above) 2 of the solutions leaving 0.381969 to be the height. –  E.O. Mar 3 '12 at 12:43
@Nuthinking: First, a not so good way. Let $w=1/h$, clear denominators. You get a cubic in $w$, with no $w^2$ term. Then use the Trigonometric Solution to this kind of cubic. However, the sensible way is to use a numerical equation-solving procedure, such as the Newton Method. Alternately, some scientific calculators have a "solve" button. –  André Nicolas Mar 3 '12 at 12:55
@EmileOkada Thanks a lot for the solution. This is what I composed: Hopefully looks about right! :) –  Nuthinking Mar 3 '12 at 13:13
@AndréNicolas I struggle to understand what you are proposing (I am too thick :(). Which app would you recommend on a mac or online? –  Nuthinking Mar 3 '12 at 13:16

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