Imagine a point \(\displaystyle (r, \theta)\) in polar coordinates. Close to that is the point \(\displaystyle (r, \theta+ \Delta\theta)\) where \(\displaystyle \Delta\theta\) is a slight change in the angle \(\displaystyle \theta\). Draw the circular arc, with center at the origin, radius r, through those two points. Draw the lines through the origin through both of those points. Finally, draw another arc, with center at the origin, with radius \(\displaystyle r+ \Delta r\) where \(\displaystyle \Delta r\) is a slight increase in r. That will intersect the two radial line at \(\displaystyle (r+\Delta r, \theta)\) and \(\displaystyle (r+ \Delta r, \theta+ \Delta\theta)\).
That is roughly a rectangle. Two sides of the "rectangle" have length \(\displaystyle \Delta r\). Since the length of a circular arc, with radius r, subtending angle \(\displaystyle \phi\), is \(\displaystyle r\phi\), the lengths of the other two sides are \(\displaystyle r\Delta\theta\) and \(\displaystyle (r+ \Delta r)\Delta\theta\). Those are "almost the same" so we can use the simpler, \(\displaystyle r\Delta\theta\), as the approximate height of the "rectangle". The area of that "rectangle" is approximately \(\displaystyle (r\Delta\theta)(\Delta r)= r\Delta\theta\Delta r\). Now, take the limit. The difference between those to "sides" disappears in the limit. \(\displaystyle \Delta r\) goes to "\(\displaystyle dr\)" and \(\displaystyle \Delta\theta\) goes to "\(\displaystyle d\theta\)" so that the "differential of area" in polar coordinates is \(\displaystyle r dr d\theta\).
