In order to get the arc length we must have $r$ as a function of $\theta$. But how can I find arc length of curve which is drawn between $r$ and $\theta$ if $r$ is an implicit function of $\theta$ like this:
\begin{align} \theta = \frac{a\cdot \left( b - \sqrt{r^2 - rf+p^2} \right) }{r} \end{align}
Where $f$ and $p$ are constants.
Since $ds^2=dr^2+r^2d\theta^2$, when $\theta$ is given as a function of $r$ the arc length is$$\int ds=\int\sqrt{1+r^2(d\theta/dr)^2}\;dr.$$In this case,$$\begin{align}r\frac{d\theta}{dr}&=-\frac{a}{r}\left(b+\frac{fr/2-p^2}{\sqrt{r^2-fr+p^2}}\right)\\\implies\int ds&=\int\sqrt{1+\frac{a^2}{r^2}\left(b^2+\frac{(fr/2-p^2)^2}{r^2-fr+p^2}+\frac{b(fr-2p^2)}{\sqrt{r^2-fr+p^2}}\right)}\;dr.\end{align}$$However, I suspect you may need to evaluate such an arc length numerically.
