I'm designing a child's toy consisting of a closed box with a hole on top; a unit tetrahedron must fit through this hole.
What is the smallest possible area of the hole?
Currently my hole is an isosceles triangle with base $1$ and height $\frac{\sqrt{2}}{2}$ (the distance between the midpoints of two opposite edges), which gives area $\frac{\sqrt{2}}{4}$. With the correct orientation, the tetrahedron is dropped straight through the hole.
Is it possible to do better considering rotations of the tetrahedra during insertion?