A square of side length $2$ has a regular octagon of side length $2$ constructed externally on each side. Let $A,B,C,D$ be the centers of the four octagons, listed in order around the square. What is the area of square $ABCD$?

1. The center of each octagon is $1+\frac{2}{2\tan 22.5^\circ}=2+\sqrt2$ units from the center of the square.
2. Adjacent centers are $90^\circ$ apart on a circle of radius $2+\sqrt2$, so a side of square $ABCD$ has length $\sqrt2(2+\sqrt2)$.
3. Therefore $[ABCD]=\bigl(\sqrt2(2+\sqrt2)\bigr)^2=\boxed{12+8\sqrt2}$.