It is mentioned in the Wikipedia page that we can define, in the complex projective line $\mathbb{CP}^1$, the following metric: $$ds^2 = \left(\frac{2}{1+|\zeta|^2}\right)^2 |d\zeta|^2 = \frac{4}{(1+|\zeta|^2)^2} d\zeta d\bar\zeta.$$
They mention that this metric must be isometric to the sphere of radius $1/\sqrt K$ in $\mathbb R^3$.
How is this metric derived? From the comment I guess it might be derivable from the metric in $S^2$, but I don't really understand how to go from there to a metric for $\mathbb{CP}^1$. Should I somehow derive it via pullback from a metric in $\mathbb C^2$? I'm not sure where to start in doing this.