Matsuyama Paradox

Seen at the Futility Closet: The seeming paradox is that you can rotate the corner quadrilaterals and make a new square that's the same size - without the middle piece. The illusion is only good for small center squares, of course. Where does the area 'go'? I made this to play around with the paradox and was surprised. Each different size center square has a specific angle at which the corner pieces make a square without a gap. What determines that angle?
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