Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$
1. **Determine the radii of the circles:**
   - Circle \( C_1 \) is centered at \( A \) and passes through \( B \). Since \( AB = 1 \), the radius of \( C_1 \) is \( 1 \).
   - Circle \( C_2 \) is centered at \( A \) and passes through \( C \). Since \( AC = 2 \), the radius of \( C_2 \) is \( 2 \).

2. **Calculate the areas of the circles:**
   - The area of circle \( C_1 \) with radius \( 1 \) is:
     \[
     \text{Area of } C_1 = \pi \times (1)^2 = \pi
     \]
   - The area of circle \( C_2 \) with radius \( 2 \) is:
     \[
     \text{Area of } C_2 = \pi \times (2)^2 = 4\pi
     \]

3. **Find the area of the region outside \( C_1 \) but inside \( C_2 \):**
   - The area outside \( C_1 \) but inside \( C_2 \) is the difference between the area of \( C_2 \) and the area of \( C_1 \):
     \[
     \text{Area outside } C_1 \text{ but inside } C_2 = 4\pi - \pi = 3\pi
     \]

The final answer is \(\boxed{3\pi}\).