An Area Worth Exploring

Geometry Level 2

It's common knowledge that the graph of x2+y21x^2 + y^2 \leq 1 is a unit disk, and the area of the region is π.\pi.

However, it's not common for someone to know the shape of the graph x2+y2=1, \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1, where \lfloor \cdot \rfloor is the floor function. To three decimal places, what is the area of the region on the coordinate plane that satisfies the equation x2+y2=1? \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1?


Note: Try to draw a hand sketch of this curve and provide analysis instead of using any software!

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