We're looking at the smaller triangle at the top of the cone. It's isosceles, so if we drop a perpendicular from apex A onto the base, we split the triangle into two back-to-back right-angled triangles. The angle at A is bisected, so we have A/2. The tangent of A/2 is half the base, which represents the smaller radius of the cone, and is d1/2 (0.9625), divided by the height of the smaller triangle we just calculated (11.2669). This is what I meant by the geometry of the triangle. We look up the angle whose tangent is the result of that division, that gives us A/2, and we simply double it to get the apex angle, A, of the cone. I hope that's clear now. I've not used the solid angle because I thought the normal presentation of the angle would be more understandable, rather like looking at the vertical cross-section of the cone represented by the nested isosceles triangles.
I rounded up the decimals, but my calculations contained more decimal places, so you may find the fourth dec place is not always what you expect (it could be out by 1 either way).