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Inside Diameter Of A Torus Rate Topic: -----

#1 User is offline   Little Bang 

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Posted 09 November 2010 - 07:47 AM

If I have a tube with diameter X what is the smallest torus I could produce?
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#2 User is offline   sanctus 

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Posted 09 November 2010 - 08:54 AM

If the length of the tube is not specified I would say infinitely small ;-)
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#3 User is offline   modest 

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Posted 09 November 2010 - 09:36 AM

View PostLittle Bang, on 09 November 2010 - 07:47 AM, said:

If I have a tube with diameter X what is the smallest torus I could produce?


The volume of a torus is V=2 \pi^2 R r^2 where R is the distance from the center of the tube to the center of the torus and r is the radius of the tube. I think the smallest torus would be where R \to r meaning the volume would be V=2 \pi^2 r^3 (where r is half of X given in the OP).

Intuitively, I think that would be right because the volume would be the same as a cylinder of height 2 \pi r and a cylinder's volume \pi r^2 h which would give V = \pi r^2 h = \pi r^2 (2 \pi r) = 2 \pi^2 r^3.

This is assuming that you mean a torus with an inner radius of zero (ie the size of the hole in the center goes to zero) and an outer radius of 2r.

The surface area would be 4 \pi^2 r^2.

~modest

EDIT: I'm sorry, I just noticed that the tread's title is "inside diameter of a torus". Sanctus' answer suddenly makes sense to me—that the inner diameter is infinitely small.
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#4 User is offline   Little Bang 

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Posted 09 November 2010 - 11:06 AM

Modest, I think you both gave basically the same answer but the detail you gave is what I was looking to find, Thank you.
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#5 User is offline   sanctus 

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Posted 10 November 2010 - 02:35 AM

But little Bang, you do not need the math for it. Sorry, yesterday I was tired that is why I did not elaborate. Just imagine a tube, if you just bend it to make a torus, you will get a torus under the condition that the tube is long enough (ok for this "long enough" you would need math). So if you have a tube which is exactly long enough then you will have an inner radius of zero.
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