Expert: Paul Klarreich - 1/29/2008

Question
I am looking for an equation that would calculate the volume of a part of a Torus. I such a way as the torus is cut in the middle to look like a wedding ring. (The flat section is against the finger and the rounded part is showing). I am interested to know the volume of the large part (or outside section) i have done tests with donuts and found that the ratio is about 60% for the large portion and 40% for the small one.( But I would like to be more precise)
The reason is that I am a glassblower and I wish to calculate the volume of glass necessary to make a bead (reinforcement) that would look like half of a torus above the outside wall at the end of a glass tube. Your help would be very useful for the members of the American scientific glassblower society (www.ASGS-glass.org)

Answer
Questioner:   Georges
Category:  Calculus
Private:  No
 
Subject:  Part of torus volume
Question:  I am looking for an equation that would calculate the volume of a part of a Torus. I such a way as the torus is cut in the middle to look like a wedding ring. (The flat section is against the finger and the rounded part is showing). I am interested to know the volume of the large part (or outside section) i have done tests with donuts and found that the ratio is about 60% for the large portion and 40% for the small one.( But I would like to be more precise)
The reason is that I am a glassblower and I wish to calculate the volume of glass necessary to make a bead (reinforcement) that would look like half of a torus above the outside wall at the end of a glass tube. Your help would be very useful for the members of the American scientific glassblower society (www.ASGS-glass.org)
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Hi, Georges,

Here's what you have to do.  I am not going to complete the computation, because the site cannot be used for commercial purposes.

1. Assume your torus is the result of taking a circle whose radius is r and whose center is at (R,0), then revolving this about the origin.  The resulting torus has a 'thickness' of 2r and a center hole of radius R-r.  The volume of the torus is the product of the area of the circle --  pi r^2 -- times the distance traveled by the center, which is also the centROID of the circle.  This center is at (R,0) and travels  a distance of 2 pi R.  So the volume is:   2 pi^2 r^2 R.

2. Now take the outer semicircle -- center still at the point (R,0), and whose radius is r.  You must now find the CENTROID of this semicircle -- check any calculus text on applications of integration.  This centroid will be at some point (X0,0).  The volume of your 'wedding ring' will be:

V = (area of semicircle) * 2 pi X0.

The area of the semicircle is, of course, 1/2 pi r^2.

So all that's left is that centroid calculation.  Your nephew taking  A.P. calculus can do that for you.