A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of square $ACEG$?

1. If the octagon has side $s$, its area is $2(1+\sqrt2)s^2$, and $AC$ (a ``short'' diagonal, skipping one vertex) has length $s\sqrt{2+\sqrt2}$, so $[ACEG]=s^2(2+\sqrt2)$.
2. Thus $\dfrac{[ACEG]}{[ABCDEFGH]}=\dfrac{2+\sqrt2}{2(1+\sqrt2)}=\dfrac{\sqrt2}2$ (rationalizing).
3. If $[ABCDEFGH]=1$, then $[ACEG]=\boxed{\dfrac{\sqrt2}2}$.