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November 17th, 2014, 11:30 AM   #1
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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?
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November 17th, 2014, 11:40 AM   #2
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Quote:
Originally Posted by slna111 View Post
I need to find the height of a spherical cap
Then find it.
Quote:
Originally Posted by slna111 View Post
Can anyone help me find the formula for that?
Yes.
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, given the volume of the cap and the radius of the sphere. (Not the base radius.)
What do you mean by "(Not the base radius)" ? Give an example.
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November 17th, 2014, 11:48 AM   #3
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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.
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November 17th, 2014, 12:59 PM   #4
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Did you mean something like this ?
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File Type: jpg Untitled.jpg (30.9 KB, 5 views)
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November 17th, 2014, 01:34 PM   #5
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Close. It would be like my attached, with the volume and R given.
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November 17th, 2014, 03:05 PM   #6
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The formula for the volume of a spherical cap in terms of the height of the cap and the radius of the sphere is

$\displaystyle V=\frac{\pi h^2}{3}(3r-h)$

Rearranging gives

$\displaystyle h^3-3h^2r+\frac{3V}{\pi}=0$

which can be difficult to solve. You may wish to consider numerical methods for known values of $V$ and $r$.
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November 17th, 2014, 05:07 PM   #7
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....easy (p = pi):
solve p*h^3-3*p*r*h^2+3*v=0 for h - Wolfram|Alpha
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November 18th, 2014, 02:32 AM   #8
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Thank you.
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November 18th, 2014, 05:15 AM   #9
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$\displaystyle h^3-3h^2r+\frac{3V}{\pi}=0$

That huge-looking 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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