A circle of radius $1$ and a square have the same centre and the same area. The length of chord $AB$ is

The area of the circle is pi * r^2 = pi * 1^2 = pi. Since the square has the same area, its side length is sqrt(pi) and half of this side is sqrt(pi)/2. The distance d from the center to the midpoint of the chord along the side of the square is sqrt(pi)/2. Using the perpendicular distance formula for a chord, half the chord length is sqrt(r^2 - d^2) = sqrt(1 - pi/4). The full length of chord AB is 2 * sqrt(1 - pi/4), which simplifies exactly to sqrt(4 - pi).

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