2020 AMC 12A 第 14 题

$ACEG$ is a square inscribed in the octagon

Place the octagon symmetrically and compute both areas; the four corner triangles make up the difference

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.