Metric properties

The rectangular box $ABCD{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ has edges of lengths $|AB|=4\sqrt{3}\mathrm{cm}$ and $|BC|=8\mathrm{cm}$. The point $S$ is the center of the lateral face $AD{D}^{\prime }{A}^{\prime }$ (see the picture) and the length of the line segment $B^{\prime }S$ is $10\mathrm{cm}$. Find the distance between the points $A$ and $A^{\prime }$.

The base $ABCD$ of a square pyramid $ABCDV$ has an edge of $4\mathrm{cm}$. The height of the pyramid is $6\mathrm{cm}$. Find the distance between the point $A$ and the point $S_{VB}$, where $S_{VB}$ is the midpoint of the edge $VB$.

The base of a pyramid is a square with the side of $4\mathrm{cm}$. The height of the pyramid is $8\mathrm{cm}$. Find the angle between the lateral edge of the pyramid and the base plane. Round your result to two decimal places.