2020 AMC 12A Problem 14

The four vertices $A, C, E, G$ form a square, since they are every other vertex of the regular octagon. Taking a unit circumradius, the octagon’s area is $2\sqrt2$ and the square $ACEG$ has diagonal equal to the circle’s diameter, giving area $2.$ The ratio is $\dfrac{2}{2\sqrt2} = \dfrac{1}{\sqrt2} = \dfrac{\sqrt2}{2}.$ Thus, B is the correct answer.