Flat earth geometry is pretty simple:

Globe earth geometry is a little less straightforward:

For a globe of r=3959 miles, the tilt angle between my perpendicular and perpendicular at the location of solar noon (sun's location) at the time of my measurement would have been ≈33.67°.

(arc distance/earth circumference)*360°

or

(2329/24901)*360°

That plus the observed elevation angle above horizontal of 55.87° leaves "gap" of about 0.46° to perpendicular.

But that's not acute enough a sun supposedly 93,000,000 miles away. The "gap" angle between tilt and observed elevated angle should be more like 0.002°, which is too precise for the kind of measuring any of us can due with poles and measuring tape. In a more ideal measurement setting, I should have measured a shadow of around 46 1/4" or so. As it was, my measurement would put the sun only 290,000 miles away from a globe earth.