Take a look at this curve.
Imagine that I have drawn this on a paper, and that I want to find the area of it. (The thickness of the line is assumed to be constant).
Assume that I use a string to trace the edge of the figure, then compare that to a ruler. Then I measure the height using the same method. I will get values for the width and height, so can I just say Area = Width $\times$ Height? I will be extending that curvy shape into a rectangle. So in this sense, could I not assume that the area is like that of a rectangle, namely Area = Width $\times$ Height?
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EDIT
If this is applicable, I'm really interested to know if this also works with a curvy shape that has non-constant thickness.
If you measure the length in the middle of the (thickened) line, then indeed the area is just this length times thickness - at least as long as the curve is smooth enough and does not bend back on itself.