When a square (or any rectangle) is inscribed in a circle, the diagonal
of the square (or rectangle) is a diameter of the circle.
Diagonal of this square = diameter = 2 x radius = 4√2 inches.
But the diagonal of a square is the hypotenuse of the right triangle that's
formed by cutting the square in half corner-to-corner.
So (diagonal)² = (side)² + (side)²
2 (side)² = (diameter)²
2 (side)² = (4√2)²
2 (side)² = 16 x 2 = 32
(side)² = 16
side = 4 inches