A simple geometric math puzzle (too simple for MO). Seeing two overlapping square beermats, I had the idea: One mat is the other rotated by an angle. All colored areas are the same (say 1, arbitrary unit). What is the size of the square and the rotation angle?

Bonus if you can solve it in your head even without the pic. (Which is suggestive; the suggestion is correct.)

Superbonus if you do the same question with two circular mats.