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/sci/ - Science & Math

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11547740 No.11547740 [Reply] [Original]

Without looking it up, derive a formula for the volume of a torus.

>> No.11547747

>>11547740

without attempting to act like you are not a faggot, describe the taste of cum

>> No.11547753

>>11547747
A bit bitter, a bit salty, like a less-strong snot taste

>> No.11547940

>>11547740
Recall the parametrisation

[math]\vec{r}(\theta, \phi)=((a \cos \theta+b) \cos \phi,(a \cos \theta+b) \sin \phi, a \sin \theta)[/math]

Use this to calculate the metric tensor using coordinate vector fields [math]\frac{\partial}{\partial \theta}, \frac{\partial}{\partial \phi}[/math] and then immerse into [math]\mathbb{R}^3[/math] to calculate the first fundamental form [math]\mathbb{I}[/math]

Then use the Hodge-star operator calculate the Riemannian volume form and integrate the constant function [math]1dV_g[/math]

The calculation of the integral is left as an exercise

>> No.11547942

The flat torus has 2-volume 1.

>> No.11547948

just cut and straighten it and it turns into a cylinder with volume 2*Pi^2*r^2*R

>> No.11547963

>>11547940
>left as an exercise
Kek

>> No.11548159

>>11547948
thats what I was thinking. though I wonder if because the interior has a smaller circumference than the outer circumference that it wouldnt be able to be "straightened" without changing its form thus is volume.

>> No.11548302

>>11547740
V1 - V2 = VT
Where V1 is the volume of water before dunking the torus in the bucket, V2 is the volume after doing so, and VT is the volume of the torus. Error includes water being spilled due to jostling, water sticking to the torus, and limited precision tools.

>> No.11548317

>>11548159
Cut it into wedges, then I think you just have to check that you can approximate those wedges as cylinders.

>> No.11548381 [DELETED] 

>>11547740
Not a math or geometry fag, but wouldn't this require that you know whatever the constant value between a Torus of X size and a perfect sphere of X size.

Since you'd probably deduce it, as a sphere and subtract what's missing...

Or, you cut the torus on one side, to make it a cylinder, then compute volume with the cylinder formula.

Again, just need to know the radius or diameter of the ring, and the diameter of the full torus.

Wouldn't be too bad.

I would actually choose the cylinder method.

>> No.11548427

>>11547740
Whats a torus?

>> No.11548497

>>11547740

This follows from Pappus's centroid theorem: the volume of the torus is equal to the surface of the plane figure - a circle - times the traveled distance of said circle's centroid C.

Surface of the plane figure - a circle - is (πr2) where r is the circle's radius i.e. from C out to the torus's edge.
Traveled distance is the circumference (2πR) where R is the radius from C to the center of the torus.

Final formula: (πr2)(2πR)

(And seriously fuck off if you expect someone to reproduce the proof for the theorem from memory.)

>> No.11548504

>>11548497
> (πr2)

(πr^2) obviously.

Fuck the crappy board software folding the Unicode super-script 2 into the normal one.

>> No.11548514

>>11547740
Area of the cross section multiplied to get the rotation.

>> No.11548548

>>11547740
Some gay shit involving solids in revolution.

>> No.11548573

>>11548504
Just use LaTeX, faggot.

>> No.11548804

>>11548497
>Pappus's centroid theorem
That's neat

>> No.11548831

>>11547740
a = radius of inner circle cross-section where it is largest
b = radius of entire torus cross section where it is largest
(((b-a)/2)^2 * pi) * ((a+((b-a)/2))*2)*pi