Rectangle $ABCD$ has $AB=4$ and $BC=2$. An ellipse passes through $A$ and $C$ and has foci at $B$ and $D$. What is the area of the ellipse? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.)

1. $A$ is on the ellipse with foci $B$ and $D$, so $2a=AB+AD=4+2=6$ and $a=3$.
2. The distance between the foci is the diagonal $BD=\sqrt{4^2+2^2}=2\sqrt5$, so $c=\sqrt5$.
3. $b^2=a^2-c^2=9-5=4$, so $b=2$.
4. The area is $\pi ab=\boxed{6\pi}$.