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

its diameter is 16 inches,

So, according to the Pythagorean Theorem, for some right triangle with legs of length a, b, and hypotenuse of length c,

then a² + b² = c²

c = 16 inches, and a = b.

a² + a² = 16²

2a² = 256

a² = 128

√a² = √128

a = √128

a = √ (64•2)

a = 8√2

So, the size is an 8√2 by 8√2 square.