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how to calculate area of horizontal band of a torus surface?

I have a ring torus of the following proportions illustrated in the diagram below.

The diagram shows a cross section of the torus (it is not proportional)

Inner hole radius = a = 5.920118575210524

tube radius = r = 82.983609982018226

I am interested in surface areas.

I know how to calculate the total surface area of the torus.

But how can I calculate the area of any horizontal strip of the surface?

I have a specific example in mind, although I would like to know the methodology too.

In the diagram there is a strip labelled d, which runs from the inner equator to a point which is c degrees above.

I have worked out that:

d=15

Therefore c = 10.3567040

How can I calculate the surface area of the torus that this strip ‘d’ covers as it rotates 360 degrees around the y axis?

I will appreciate any help anyone can give.

Thanks,

Tom

Attachment 23437

Re: how to calculate area of horizontal band of a torus surface?

Quote:

Originally Posted by

**tombarnett** I have a ring torus of the following proportions illustrated in the diagram below.

The diagram shows a cross section of the torus (it is not proportional)

Inner hole radius = a = 5.920118575210524

tube radius = r = 82.983609982018226

I am interested in surface areas.

I know how to calculate the total surface area of the torus.

But how can I calculate the area of any horizontal strip of the surface?

I have a specific example in mind, although I would like to know the methodology too.

In the diagram there is a strip labelled d, which runs from the inner equator to a point which is c degrees above.

I have worked out that:

d=15

Therefore c = 10.3567040

How can I calculate the surface area of the torus that this strip ‘d’ covers as it rotates 360 degrees around the y axis?

I will appreciate any help anyone can give.

Thanks,

Tom

Have a look here: Pappus's centroid theorem - Wikipedia, the free encyclopedia

(Remark: I'm not sure if this method works in your case properly ... (Thinking))

Re: how to calculate area of horizontal band of a torus surface?

Thanks for this - it has been very useful.

I now understand I could follow the first theorem. Now the question is though - how can I find the geometric centroid of the curve I have described?

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Re: how to calculate area of horizontal band of a torus surface?

Quote:

Originally Posted by

**tombarnett** Thanks for this - it has been very useful.

I now understand I could follow the first theorem. Now the question is though - how can I find the geometric centroid of the curve I have described?

1. Let $\displaystyle \rho$ denotes the radius of the circle which is described by the centroid of d during the rotation. Then

$\displaystyle \rho = a + \left(r - r \cdot \cos \left(\frac c2 \right) \right)$

3. The area is calculated by:

$\displaystyle A = 2 \pi \rho \cdot d$

4. I've got $\displaystyle A \approx 589.87897$

Re: how to calculate area of horizontal band of a torus surface?

Quote:

Originally Posted by

**earboth** 1. Let $\displaystyle \rho$ denotes the radius of the circle which is described by the centroid of d during the rotation. Then

$\displaystyle \rho = a + \left(r - r \cdot \cos \left(\frac c2 \right) \right)$

3. The area is calculated by:

$\displaystyle A = 2 \pi \rho \cdot d$

4. I've got $\displaystyle A \approx 589.87897$

Brilliant!

So the centroid *is* located at the point intersected by half the angle. That was the part that was confusing me. and so I got the same answer.

Thank you so much for taking the time to help me out here. Its very kind of you. I am now out of my mathematical impass and can continue on the journey!

Respect.

Tom