How do discretize the following region?

Question

Consider the following region:

reg = Polyhedron[{{-0.8420454545454543, -1.8784090909090905, 
   37.99999999999999}, {-1.95, -4.35, 88.}, {-1.95, 4.35, 
   88.}, {-0.8420454545454545, 1.8784090909090907, 
   38.}, {1.95, -4.35, 88.}, {1.95, 4.35, 
   88.}, {0.8420454545454543, -1.8784090909090905, 
   37.99999999999999}, {0.8420454545454543, 1.8784090909090905, 
   37.99999999999999}}, 
    {{1, 2, 3, 4}, {5, 6, 3, 2}, {7, 5, 2, 1}, {8, 6, 5, 7}, {4, 3, 
   6, 8}, {8, 7, 1, 4}}];
Graphics3D[reg]

I need to define the minimal and maximal polar angles $\theta = \arccos(z/\sqrt{x^{2}+y^{2}+z^{2}})$ for this region. (well, the minimal angle is clearly 0 in this case, but I would like to automatize it)

For this purpose, I want to use the answer to this question. It requires the discretization of (the boundary of) the region, i.e.

DiscretizeRegion[reg]

or

BoundaryDiscretizeRegion[reg]

However, I get the following message:

A non-degenerate region is expected at position 1 of DiscretizeRegion

Could you please tell me how either to discretize it properly or to evaluate minimal and maximal polar angles automatically?

Answer 1

This looks like a bug, which has probably been fixed in either version 13.1 or 13.2.

A workaround is to manually specify the bounds, for example:

BoundaryDiscretizeRegion[reg, {{-100, 100}, {-100, 100}, {-100, 100}}]