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

1. ## 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

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

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 ... )

3. ## 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?

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

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

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

3. The area is calculated by:

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

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

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

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

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

3. The area is calculated by:

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

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