What is the height of a circular cone with surface line ("Mantellinie") s, which has...

The "Mantellinie" $s$ is what you think it is: It's the distance from the tip of the cone to the rim at the bottom. Given $s$, you are free to choose the height h in the interval $[0,s]$, and the radius of the bottom circle will then be $r=\sqrt{s^2-h^2}$. Therefore we have to maximize the function
$V(h)=\frac{\pi }{3}{r}^{2}h=\frac{\pi }{3}(s^{2}h-h^3)(0\le h\le s).$
As $V(0)=V(s)=0$ this maximum is taken for some $h\in ]0,s[$ , which will come to the fore by solving $V^{\prime }(h)=\frac{\pi }{3}(s^2-3h^2)=0$ for $h$. There is only one solution in the interval $]0,s[$ , namely
$h=\frac{s}{\sqrt{3}},$
and this is the height you wanted to know.