Prove volume of a torus equation

In summary, the formula for volume of a torus can be found by slicing it into horizontal rings or vertical discs and integrating. Any cross-section shape can be used, such as an ellipse or polygon, and the x and y positions need to be known. Pappus' centroid theorem can also be used to find the volume of revolution.
  • #1
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how does the formula for volume of a torus work.
is there a proof with integration??

could you use an ellipse?
 
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  • #2
It's a surface of revolution, so yes, there is a formula (method of cylindrical shells).
 
  • #3
brandy said:
how does the formula for volume of a torus work.
is there a proof with integration??

Hi brandy! :smile:

Yes, slice it into horizontal rings (or vertical discs), and integrate. :wink:
could you use an ellipse?

uhhh? :confused:
 
  • #4
by ellipse i mean, an ellipse is revolved around a circular ring.
sort of like taking a torus and stretching it upward...
get me?
 
  • #5
also what could you use a shape like a polygon or something and revolve it around a circle.

how could you would out the volume?
 
  • #6
brandy said:
by ellipse i mean, an ellipse is revolved around a circular ring.
sort of like taking a torus and stretching it upward...
get me?
brandy said:
also what could you use a shape like a polygon or something and revolve it around a circle.

how could you would out the volume?

ah! got you! :smile:

yes, you can use any cross-section shape …

if you slice it into horizontal rings, you just need to know the width of the polygon at each height …

and if you use vertical discs, you just need to know the height of the polygon at each width :wink:
 
  • #7
sorry. i kept rewording what i was going to say and i didnt read what i wrote.
so all you need is the x,y positions ?
can you explain how this works??
keep in mind i know nothing AT ALL.
 
  • #8
i mean like wat do u do to the points... to get the volume of revolution
 
  • #9
Have you considered using Pappus' centroid theorem?
 

What is the equation for finding the volume of a torus?

The equation for finding the volume of a torus is V = π^2 * r^2 * R, where r is the radius of the torus and R is the distance from the center of the torus to the center of the tube.

How do you prove the volume of a torus equation?

The volume of a torus can be proven using calculus and integration. By slicing the torus into infinitely thin rings and calculating the volume of each ring, the total volume of the torus can be found.

What is the significance of the π^2 term in the volume of a torus equation?

The π^2 term in the volume of a torus equation represents the area of a circle (π * r^2) being multiplied by the circumference of the torus (2πR). This combines the two-dimensional area with the one-dimensional circumference to calculate the three-dimensional volume.

Can the volume of a torus be calculated using other methods?

Yes, the volume of a torus can also be calculated using geometric methods such as the Cavalieri's principle or Pappus's centroid theorem. These methods involve calculating the volume of a solid of revolution and can be used to prove the volume of a torus equation.

Does the volume of a torus equation apply to all torus shapes?

No, the volume of a torus equation only applies to a specific type of torus known as a "standard torus." This type of torus has a circular cross-section and a circular tube, with the tube centered at the origin of the torus. Other types of tori may have different equations for calculating their volume.

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