I have been stumped on this problem that is a bit convoluted to explain but is explained in more detail in this post here if curious.

But right now I am just hoping to try and find the answer myself. Basically I have a cone formed by n trapeziums that extend out at a tilt angle.

The original post provides a formula for the saw tilt angle or essentially the acute angle of the trapeziums necessary to have them join at the seams. There is only one problem, I hopped in CAD and modeled this parametrically and this calculation is not always right, it is often a sliver off.

I cant figure out how to derive my own equation. I feel like I could use linear algebra here but vector/matrix operations are really difficult to plug into my CAD software. I am trying to find a plain algebra way to calculate this angle.

Where I am stumped is the 3D nature of it all. I know that if the top of this cone was "flat" then the angle necessary to have these trapeziums join would be the polygon angle. However given the trapezium prisms extend out at a tilt angle I realistically need to "project" this polygon angle onto their plane.

In the diagram below the top row just shows it with regular cuboids with their sides and the angle needed in a flat plane above the cone.

The lower row then draws a plane on one of the faces and projects that 60% angle onto the plane of the trapezium volume. The angle between the cuboid side and that projected angle onto its plane is what I need to derive.

The biggest thing this shows for me is that this problem can likely be simpler if I consider that all necessary planes share a common point where the cuboids join. I know the angle between these two planes (at least I think I do)

I guess my issue is I am realizing I have nearly no tools to solve this without linear algebra. I estimated the correct values and the shape of the values makes it feel like there should be a solution that can be expressed in terms of trigonometric functions. Is there a discipline of geometry that helps solve problems like this? Problems where you are dealing with angles in 3D space and their projections onto planes that are at an angle to them.

I made diagrams of the situation involved but this forum does not allow pictures. The linked stack exchange math post has the diagrams. In this question I am more looking for a pointer as to what kind of mathematical tools/techniques might be employed here. A solution would be great too but I am down to do the work.