I'm not sure how you got only part of the formula... perhaps you made a calculation error? Did you remember that $x^2 + y^2 = r^2$ and not $x^2 + y^2 = r$?
At any rate, I assume that you're starting with the formula $S = \iint_D \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y}\right)^2}\,dx\,dy$ and going from there.
Naturally, you'll need to use the chain rule for partial derivatives $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial x}$ $\frac{\partial f}{\partial y} = \frac{\partial f}{\partial r}\frac{\partial r}{\partial y} + \frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial y},$ while in the process getting rid of any $x$- and $y$-terms via the formulas $x = r\cos \theta$ $y = r\sin \theta.$ Finally, the change of variables formula lets you write (formally) $dx\,dy = r\,dr\,d\theta,$ so that should account for the final $r$-term outside of the square root.
I'll leave the details of the calculation up to you.