Consider the function $r=f(\theta )$ in polar coordinates. The length of an arc of a circle is just

$S=\theta r$

Where $r$ is the radius of the circle and $\theta $ is the angle that represents this arc. But since $r=f(\theta )$ and $\theta $ Should approach zero so that we can get the exact value of the arc, So

$\phantom{\rule{thinmathspace}{0ex}}dS=f(\theta )\phantom{\rule{thinmathspace}{0ex}}d\theta $

Integrating from ${\theta}_{1}$ to ${\theta}_{2}$, we get :

$S={\int}_{{\theta}_{1}}^{{\theta}_{2}}f(\theta )\phantom{\rule{thinmathspace}{0ex}}d\theta $

But the actual formula for the length of a curve in polar coordinates is

${\int}_{{\theta}_{1}}^{{\theta}_{2}}\sqrt{{f}^{2}(\theta )+{f}^{\prime}(\theta {)}^{2}}\phantom{\rule{thinmathspace}{0ex}}d\theta .$

I know that my approach isn’t rigorous enough, but it’s is still reasonable, so why it is different from the actual formula?