The figure shows $7$ circles of radius $2$ within a larger circle: one at the center and six around it, each tangent to the center circle and to its two neighbors, with all outer circles tangent to the large circle. All intersections occur at points of tangency. What is the area of the region inside the larger circle but outside all $7$ circles of radius $2$?

1. Each outer circle's center is at distance $2r$ from the center (tangent to the center circle), so the large circle's radius is $R=2r+r=3r$.
2. Area $=\pi R^2-7\pi r^2=\pi(3r)^2-7\pi r^2=9\pi r^2-7\pi r^2=2\pi r^2$.
3. With $r=2$: area $=2\pi(4)=\boxed{8\pi}$.