What are the equations for surface area and volume of a torus,
*including* torii with minor>major radius?

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I don't remember that calculus stuff very well, but since no one's 
helped you yet, me and my pocket calculator had a go at it:

volume:
  a:=2*int( sqrt(r^2-(x-R)^2) ,x,0,R+r )  // 1/2 torus cross-section
  int( a*R,theta,0,2*pi )                 // object of revolution
> R*pi*( pi*r*abs(r)+2*R*sqrt(r^2-R^2)+2*pi*asin(R*abs(1/r))/180*r^2 )

Odds are this is wrong.. Not sure about line 2.

area:
  c:=2*sin(R/r)*r 			// part of arc inside torus
  int( (2*pi*r-c)*R ,theta,0,2*pi )
> -4*r*R*pi*(sin(R/r)-pi)

or something

sig.


David Fontaine wrote:
> 
> What are the equations for surface area and volume of a torus,
> *including* torii with minor>major radius?
> 
> --
> Homepage: http://www.faricy.net/~davidf/
> ___     ______________________________
>  | \     |_       <dav### [at] faricynet>
>  |_/avid |ontaine      <ICQ 55354965>

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Darn. Both are wrong. Setting R=r should be equal to a sphere, but it's
not:
A!=-4*pi*r^2( (sin(1)-pi )
V!=2*pi^2*r^3



Sigmund Kyrre Aas wrote:
> 
> I don't remember that calculus stuff very well, but since no one's
> helped you yet, me and my pocket calculator had a go at it:
> 
> volume:
>   a:=2*int( sqrt(r^2-(x-R)^2) ,x,0,R+r )  // 1/2 torus cross-section
>   int( a*R,theta,0,2*pi )                 // object of revolution
> > R*pi*( pi*r*abs(r)+2*R*sqrt(r^2-R^2)+2*pi*asin(R*abs(1/r))/180*r^2 )
> 
> Odds are this is wrong.. Not sure about line 2.
> 
> area:
>   c:=2*sin(R/r)*r                       // part of arc inside torus
>   int( (2*pi*r-c)*R ,theta,0,2*pi )
> > -4*r*R*pi*(sin(R/r)-pi)
> 
> or something
> 
> sig.
> 
> David Fontaine wrote:
> >
> > What are the equations for surface area and volume of a torus,
> > *including* torii with minor>major radius?
> >
> > --
> > Homepage: http://www.faricy.net/~davidf/
> > ___     ______________________________
> >  | \     |_       <dav### [at] faricynet>
> >  |_/avid |ontaine      <ICQ 55354965>

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It's easy enough for normal torii, but spindle torii are a pain in the arse!
Well I don't need it that badly, I can work it out another way.

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Homepage: http://www.faricy.net/~davidf/
___     ______________________________
 | \     |_       <dav### [at] faricynet>
 |_/avid |ontaine      <ICQ 55354965>

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David Fontaine wrote:

> What are the equations for surface area and volume of a torus,
> *including* torii with minor>major radius?
>

http://mathworld.wolfram.com/T/Torus.html

Haven't checked it, but the formulas there probably won't
take r > R into consideration, so you will have to subtract
the overlapping areas/volumes.

--
Gerald
ger### [at] aonat

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> Haven't checked it, but the formulas there probably won't
> take r > R into consideration, so you will have to subtract
> the overlapping areas/volumes.

It's more complicated than that, because with major=0 you should get a
sphere but the only equations I've seen return 0 for both.

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Homepage: http://www.faricy.net/~davidf/
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 | \     |_       <dav### [at] faricynet>
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Well, I had this whole big thing typed out explaining step by step how I
found the formulas (so you could solve similar problems by yourself),
but Netscape crashes and I lost everything, so I'm just going to give
you the answers rather than type it all out again.

These formulas only work for a spindle tours.  r1 is the major radius
and r2 is the minor radius.

V = 4/3*pi*(r2^2-r1^2)^(3/2)

A = 4*pi*r2*sqrt(r2^2-r1^2)

I'm pretty sure that these are correct.  I hope this helps.

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Sigmund Kyrre Aas wrote in message <38628A92.374CC2C2@stud.ntnu.no>...
>Darn. Both are wrong. Setting R=r should be equal to a sphere, but it's
>not:
>A!=-4*pi*r^2( (sin(1)-pi )
>V!=2*pi^2*r^3


Setting R=r should give you the area for two spheres, thus:
A=8*pi*r^2
V=(8/3)*pi*r^3

Mark

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> Setting R=r should give you the area for two spheres, thus:

'fraid not.

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Kevin Wampler wrote:

> I'm pretty sure that these are correct.

Well, except for a really whopping error I made they were.
I redid my calculations and now I think I have the correct formulas:

V = 4*pi*(1/6*r1^2*sqrt(r2^2-r1^2)+1/4*pi*r1*r2^2
    -1/2*r1*r2^2*asin(-r1/r2)+1/3*r2^2*sqrt(r2^2-r1^2))

A = 4*pi*(r1*r2*acos(-r1/r2)+r2*sqrt(r2^2-r1^2))

Now, if I didn't make a typo, those should work much better.

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