Polar Coordinate Question

https://preview.redd.it/578j7vvumggd1.png?width=1853&format=png&auto=webp&s=ace08487152fab9c57b553374c8a73a44309c660

Can someone please explain why these polar coordinates are not the same? Which one would be correct if I wanted to evaluate the area solely inside the green circle?

The first equation I solved for the area of the green circle and subtracted by the margin in-between both objects. The second equation I just did 1/2 int r^2 outside - r^2 inside.

Edit: I think I understand now, is it because the margin cannot be calculated accurately using the first equation? It looks identical with the naked eye but you can't just multiply it by 4 to get 2 because the structure is asymmetrical.