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I'm very math illiterate, so please be patient with me if my question sounds silly or misinformed.

I'm trying to find the height and width axis radii of an ellipsoid, when I only know its volume and its length axis radius.

The volume is $15$ mL ($15,000$ mm$^3$) and the length axis radius is $55$ mm.

The calculator on Google gives these formulas to solve for the different variables in an ellipsoid (volume ($V$) and length ($a$), width ($b$), and height ($c$) axis radii).

To solve for the volume:

$$V = \frac{4}{3}\pi abc$$

To solve for the length axis radius ($b$ and $c$ can be swapped with $a$ to solve for those radii):

$$a = 3\frac{V}{4\pi bc}$$

The calculator doesn't return anything unless I provide values for three out of four of these variables.

I've asked for help with this in a few other places to cast a wide net, but the only person to respond so far says that s/he doesn't think it's possible to solve for $b$ and $c$ individually. I don't want to give up hope just yet, though, so I'm asking here also.

I have the feeling that I should at least be able to find a range of possible values that $b$ and $c$ would need to be in order to produce an ellipsoid that has the given $V$ and $a$.

Is it possible to do this, and if so, how?

A related and embarrassing followup question:

The value of the variable $V$ is in mL/mm$^3$ and the variables $a$, $b$, and $c$ are in mm. Should I be entering $V = 15,000$ (or $15$) and $a = 55$ in the calculator for the values of those variables, or should I be converting these values to different units?

I don't even know what tags to use for this question, so I only picked conic-sections.

Nick
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  • if you have a solid of revolution, things change. Meaning two of your "radii" would be the same. The name changes also https://en.wikipedia.org/wiki/Spheroid – Will Jagy Sep 14 '18 at 00:51
  • the person who replied to me earlier edited his answer to say that I could determine $bc$, then set $b$ to any value and $c$ would be $0.651 \div b$ (because $ac = 15 \div [(\frac{4}{3})\pi 5.5]$, when converting 55mm to 5.5cm). If I used this formula to write a script that renders the different possible shapes, would this work? Or should I re ask this question with the formulas for a spheroid? – Nick Sep 14 '18 at 01:09
  • I don't think you should do anything until this is understood. How do you know the shape you have in mind is an ellipsoid? – Will Jagy Sep 14 '18 at 01:14
  • I don't think I fully understand the difference between a spheroid and an ellipsoid. The shape I have in mind is definitely a prolate Spheroid, based on the picture and descriptions from Wikipedia. – Nick Sep 14 '18 at 01:18
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    So, if you had one made, and you held it with one finger at each end of the longest axis, you could revolve it and it would not appear to change, just revolve? If so, let $a$ be the longest "radius" and take $b=c.$ So $a>b$ and $V = \frac{4}{3} \pi a b^2$ – Will Jagy Sep 14 '18 at 01:32
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    I get $b^2 \approx 65.1088$ and $b \approx 8.0690049$ millimeters. – Will Jagy Sep 14 '18 at 01:39
  • Does that seem realistic, radius of the "equator" of a long cigar shape thing about 8.07 mm? – Will Jagy Sep 14 '18 at 01:42
  • Yes definitely. Should "radius" actually be "diameter" here? And would this be the right formula to solve for $b$?: $b^2 = 3\frac{V}{4\pi a}$? – Nick Sep 14 '18 at 01:44
  • Yeah I think that sounds right. I just realized that the value I used for the volume in this question is actually twice as large as the actual volume of the thing I'm trying to find the rough size of. It should be more egg shaped than cigar shaped, but the shape only "revolves" when you spin it on the longest axis--it doesn't change at all. – Nick Sep 14 '18 at 01:46
  • if it is egg shaped, none of this is valid: it is not an ellipsoid at all – Will Jagy Sep 14 '18 at 02:03
  • when I say egg shaped, i mean it in the way that you said your calculation made a cigar shape thing. i only mean it should be shorter and fatter than a cigar -- it shouldn't be wider towards one end of the long axis and pointier towards the other end, like a real egg – Nick Sep 14 '18 at 02:07

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