The radius of a circle is 2 inches. When the radius is doubled, by how many square inches is the area increased?  Express your  answer in terms of $\pi$.
The area of the original circle is $\pi \cdot 2^2=4\pi$ square inches. After doubling the radius to $2\cdot2=4$ inches, the area increases to $\pi \cdot 4^2=16\pi$ square inches. So the increase is $16\pi-4\pi=\boxed{12\pi}$ square inches.