Fitting without collision

[Discus84]

HI,

 

 The yellow line is wall, red is a box. I would like to know if the box fits by turning and sliding into opening. I put it into a circle as a bounding box and manually moved it around to check. 

There must be a more elegant way of doing this and mabe a way to calculate the max height of such box knowing its width and the wall depth with header height.

 

Thanks!