I want to find the arc length formula for a polar curve directly without having to use the arc length formula for cartesian coordinates. The integrand $\sqrt{r^2+(r')^2}$ is a bit strange to me because it indicated a triangle with bases $r$ and $r'$ and hypotenuse infinitesimally close to $r$. For reference the formula I am thining about is $\int\sqrt{r^2+(r')^2}d\theta$ where $r=r(\theta)$.
The cartesian arc length formula makes more sense to me because all sides of the triangle $ds,dx,dy$ are infinitesimal.