What size square can be inscribed in a circle that has a radius of 8 inches?

Since it's a square, the diagonals must intersect at right angles, and since the four corners of the square must touch the circle those diagonals must be diameters of the circle. So, you'll have in effect four right triangles with adjacent sides of 8", and the third side (opposite the right angle) must be the square root of (8 squared + 8 squared), i.e., sq rt of 128, which is the length of the one side of the square.