We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle?
1. **Determine the side length of the equilateral triangle:**
   - The radius \( r \) of the inscribed circle in an equilateral triangle is given by the formula:
     \[
     r = \frac{s \sqrt{3}}{6}
     \]
     where \( s \) is the side length of the triangle.
   - Given \( r = 1 \), we can solve for \( s \):
     \[
     1 = \frac{s \sqrt{3}}{6} \implies s \sqrt{3} = 6 \implies s = \frac{6}{\sqrt{3}} = 2\sqrt{3}
     \]

2. **Calculate the area of the equilateral triangle:**
   - The area \( A \) of an equilateral triangle with side length \( s \) is given by:
     \[
     A = \frac{s^2 \sqrt{3}}{4}
     \]
   - Substituting \( s = 2\sqrt{3} \):
     \[
     A = \frac{(2\sqrt{3})^2 \sqrt{3}}{4} = \frac{4 \cdot 3 \cdot \sqrt{3}}{4} = 3\sqrt{3}
     \]

3. **Calculate the area of the inscribed circle:**
   - The area \( A_{\text{circle}} \) of a circle with radius \( r \) is given by:
     \[
     A_{\text{circle}} = \pi r^2
     \]
   - Given \( r = 1 \):
     \[
     A_{\text{circle}} = \pi \cdot 1^2 = \pi
     \]

4. **Calculate the area of the region inside the triangle but outside the circle:**
   - This area is the difference between the area of the triangle and the area of the circle:
     \[
     A_{\text{region}} = A_{\text{triangle}} - A_{\text{circle}} = 3\sqrt{3} - \pi
     \]

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