I have been struggling to create the inverse of another equation mentioned on this website: Calculate dimensions of square inside a rotated square
Relevant image:

Supposedly this solves the opposite problem but I have no idea what "s" means as "L" is what they are looking for.
$$s = \frac{L}{\cos \theta + \sin \theta},$$
I know the dimensions of the square inside and would like to know the minimum size outer square to cover the inner square. Bonus points for derivation and support for any rectangle surrounded by a square. If it makes it any easier then I will be using a fixed angle of 10 degrees but may change that so taking in an angle is better.