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 $\pi$.
To solve this problem, I need to recall the formula for the area of a circle, which is $\pi r^2$, where $r$ is the radius. I also need to understand that doubling the radius means multiplying it by 2, so the new radius is $2 \times 2 = 4$ inches. Now, I can plug in the original and the new radii into the formula and find the difference in the areas. The original area is $\pi \times 2^2 = 4\pi$ square inches. The new area is $\pi \times 4^2 = 16\pi$ square inches. The difference is $16\pi - 4\pi = 12\pi$ square inches. Therefore, the area is increased by $12\pi$ square inches when the radius is doubled. 