Least efficient method of packing unit squares into a circle?

Question

I am wondering what the least number of unit squares that can pack a circle of diameter $k$ is.

Consider a circle packed when you cannot fit another unit square inside it without it overlapping with another square or going outside the circle. These unit squares can be rotated and translated in any way, and they need not all have the same orientation.

It may be useful if there is a known method of arranging squares over the plane such that no additional squares can be fit onto the plane and the amount of the plane not covered by squares in a given area is at its maximum.

I am not sure where to begin. I have seen approaches to the related problem ‘how many squares can we fit into a circle’ but I don’t see how I could approach this problem similarly.

Answer 1

For a large number of squares, fill the center of the circle with unit squares in a staggered grid where the squares in each row are distance 1–ε apart, and the rows themselves are √2-½-ε apart, which means no other squares (shown in red) will be able to fit between them.

staggered squares