This is more obscure than other values that I baldly assert.
Consider a parallelepiped with four unit square faces, and two rhombuses faces with an angle of α.
Drop that 10<sup>6</sup> times and see how many times it falls on the corner with angle α.
I claim that it falls on one of those 4 particular corners about (((π − α)/π)8)10<sup>6</sup> times.
That corner is locally congruent with the corner of the imagined room.
4π is the whole set of directions in 3D and thus the magnitude of said corner is 4π(((π − α)/π)8) = (π − α)/2.
There are two such corners in said exercise, thus π − α.