Gaussian curvature of a cone away from its vertex
= Gaussian curvature of a cone away from its vertex
Every regular cone parametrized by $X(r,t)=r\,c(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, while
$$
X_{rr}=0,
\qquad
X_{rt}=c'(t)
$$
is tangent, so two coefficients of the <second fundamental form> vanish and $\det II=0$.