Cornering the cylinder

What is the area of the smallest square mirror that can be used to 'hide' a cylinder with a radius of $1$ and height of $2$ in a corner of the cubic room?

One side (edge) of the mirror must rest on the floor and its two upper vertices must touch the two walls, but the cylinder is free to be orientated in any way as long as it stays bounded within the corner of the room and the plane which the smallest mirror lies. If this area is $A,$ please submit $\lfloor 10^5 \cdot A \rfloor$ as the answer.

'Hide' : Totally hiding the shape from view is not the real intention here. We are only interested in the smallest square that can be inscribed in the plane that makes up the last boundary of containable(?) space.