2021 Gauss 8 Problem 23

A dog's leash is 4 m long and is attached to the corner of a 2 m \times 2 m square doghouse at C, as shown. The dog is attached to the other end of the leash, at D. What is the area outside of the doghouse in which the dog can play?

\textbf{(A)}\ 14\pi \text{ m}^2\quad \textbf{(B)}\ 16\pi \text{ m}^2\quad \textbf{(C)}\ 20\pi \text{ m}^2\quad \textbf{(D)}\ 15\pi \text{ m}^2\quad \textbf{(E)}\ 24\pi \text{ m}^2

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.