FLRW area distance versus radial proper distance

In curvature coordinates the present sphere area is $4\pi (R_{0}r{)}^2$. Its transverse area distance is $R_{0}r$, whereas radial proper distance is $R_0{\int }_0^{r}dr/\sqrt{1-K{r}^2}$. Confusing them invalidates a flat-space distance identity in curved geometry.