You can imagine drawing a line just inside the perimeter of this shape, and then a line just inside that and so on and eventually I believe you will end up with something like a "skeleton" line:

So the perimeter of the skeleton line is 40. It is basically twice the length of the line. The original perimeter is 46.28. The rule for the area is it is the average of the two perimeters times the thickness around the skeleton. The area of the shape above is (46.28+40)/2 = 43.14*1

If it were twice as thick everywhere it would be times 2 instead of times 1.

It's easy to verify that the rule works for either a circle or a square where you consider the length of the skeleton to be 0. Also it works for rectangles. I haven't yet extended it to more complicated shapes.

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