Minimum ceiling height to move a closet to upright position I brought a closet today. It has dimension $a\times b\times c$ where $c$ is the height and $a\leq b \leq c$. To assemble it, I have to lay it out on the ground, then move it to upright position. I realized if I just move it in the way in this picture, then it would require the height of the ceiling to be $\sqrt{c^2+a^2}$.
Is this height required if we consider all possible ways to move the closet? Or is there a smart way to use less height?

This is a image from IKEA's instruction that comes with the closet.

If you're treating the box as a rigid body (you can't chop it, squish it, burn it, etc.), then we can make the following statements:


*

*The box starts with a rotation angle of $\theta = 0$, and you want to rotate it to get $\theta = 90^\circ = \pi/2$.

*Because the box is rigid, it cannot be deformed; thus, its angle $\theta(t)$ must be 
continuous.

*By the Intermediate Value Theorem, the box must pass through every angle between $0$ and $90^\circ = \pi/2$.

*Thus, the box must pass through the angle where the required height is $\sqrt {a^2 + b^2}$. There is no way to "do any kind of movement to the box" such that we could "stand it up [using] less than that height."

