While studying isoperimetric problems, I ran into a solution using a differential area element based on the area of a circle sector: $$A_{sector}=\frac{1}{2}r^{2}\theta$$ for radius $r$ & angle $\theta$ subtended by the arc. The solution I saw took $r=r(\theta)$, and defined the differential area as:
$$dA=\frac{1}{2}r(\theta)^2d\theta$$ I'm confused as to how the author of the solution arrived at that definition, as I can't see a way that taking the derivative of the original area definition could produce the latter.