A regular octagon has side length $4$. At each vertex, a circular sector of radius $2$ is drawn inside the octagon, with its radii along the adjacent sides. What is the area inside the octagon but outside the eight sectors?

1. The regular octagon has area $2(1+\sqrt2)(4^2)=32(1+\sqrt2)$. Each sector has angle $135^\circ$ and radius $2$, so the eight sectors have total area $8\cdot\frac{135}{360}\pi(2^2)=12\pi$.
2. The sectors have radius $2$, half the side, so they do not overlap. The shaded area is $\boxed{32(1+\sqrt2)-12\pi}$.