$\displaystyle $$\begin{eqnarray*}L & = & \int_{\alpha}^{\beta} \sqrt{ \left\{ f(\theta) \right\}^2 + \left\{ f'(\theta) \right\}^2 } ~ d\theta\\[0.5em] & = & \int_{\alpha}^{\beta} \sqrt{ r^2 + \left( r' \right)^2 } ~ d\theta \end{eqnarray*}$$ $