Figuring an angle and an arc from the circumference and height of a cone.

CHHCFC

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I am older and I've not messed with Trig/Geometry in quite some time. I have a problem I need help solving.
I have a cone with circumference "X"/diameter "X1" at the bottom and a height of "Y". I have to wrap a "T" high collar all the way around this cone at different heights "H" from the bottom (or top).

I know the values for X, X1, Y, H and T. What I need to figure out is: 1) "C", the bottom circumference of my collar at height "H" up the cone from the bottom, 2) "A1", the angle of the cone surface from bottom to top (predicated, I assume, on the change in circumference over distance "H") and finally, 3) from "A1", figure "A2", the arc of the collar in order for it to lie flat against the surface of the cone as it's placed over the cone (without an arc, the collar will point straight up from wherever it sits on the cone like a crown sitting on your head. I need the collar to lie slanted inward towards the center of the cone so that it lies flat against the cone's surface, which requires the collar to be arc shaped, based on the value of A1 (I think)).

Obviously, the math is a bit out of my league. What I'm looking for is an equation into which I can plug my known values and get my unknowns.

See the attached illustration.

Thanks in advance for any help.
 

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hoosie

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CHHCFC

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I thought about that; couldn't I use Pythagoras (a sq'd + b sq'd = c sq'd) for that? I know M from my 2 circumferences/diameters (one minus the other) and I know T. Wouldn't that allow me to find S?
 

hoosie

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My apologies I didn’t realise you knew the diameter of the top of the cone (this was not mentioned in your initial post which is why I left it blank on my diagram).

With this measurement ( let it be r) I agree with you that you can use Pythagoras Theorem to determine s.
The difference in the two diameters (x - r) doesn’t give m directly? Why not?
 

hoosie

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CHHCFC

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Very good. Thank you for the help. I appreciate it.
As I said initially, it's been a long, long time since I messed with Trig. Let me play with what you've shown me. If I have more questions, I will post them.
As an aside, I happened to find this site on the web the other evening while working on a different problem. It has a calculator online which does almost exactly what I'm talking about above, except it gives you a template for the collar as opposed to the arc degrees as a solution. You need to know the final collar size (d, b & t) in advance to use it, but that's doable from the initial cone size with some extra math per your comments above.
https://www.onlinelabels.com/label-generator-tools/tapered-label-generator.aspx
 
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