Coin thickness for a one-in-three edge landing

Model a coin as a solid cylinder of radius $1$ and thickness $t$. To decide how it lands, enclose it in its concentric circumscribed sphere, pick a uniformly random point on the sphere, and say the coin lands on its edge if the line from the centre to that point passes through the curved side of the cylinder. Under this model, the thickness giving a $\tfrac13$ chance of landing on edge is $\tfrac1{\sqrt k}$ for an integer $k$. Find $k$.

$k=2$. By Archimedes' hat-box theorem, the sphere's surface between two parallel planes has area proportional to their separation, so the edge probability is $\tfrac{t}{2R}$ with $R=\sqrt{1+t^2/4}$. Setting $\tfrac t{2R}=\tfrac13$ gives $9t^2=4+t^2$, so $t^2=\tfrac12$ and $t=\tfrac1{\sqrt2}$.