Metric properties

Let $ABCDEF{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }{E}^{\prime }{F}^{\prime }$ be a regular hexagonal prism with the base edge length of $4\mathrm{cm}$ and the height of $8\mathrm{cm}$. Find the angle between the line $B{A}^{\prime }$ and the plane $AE{E}^{\prime }$ (see the picture). Round the result to two decimal places.

Let $ABCDEF{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }{E}^{\prime }{F}^{\prime }$ be a regular hexagonal prism with the base edge length of $4\mathrm{cm}$ and the height of $6\mathrm{cm}$. Find the distance between the lines $FA$ and $D^{\prime }{C}^{\prime }$ (see the picture).

Let $ABCD$ be a regular tetrahedron with the height of $3\sqrt{6}\mathrm{cm}$. Find the edge length of the tetrahedron (see the picture).

The side of a regular hexagonal prism $ABCDEF{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }{E}^{\prime }{F}^{\prime }$ is $a=3\mathrm{cm}$ and the height $v=8\mathrm{cm}$. Find the length of the diagonal $A{D}^{\prime }$.