Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?

1. **Assume the side length of the regular hexagon is 1.**
   - Let the side length of the hexagon be \( s = 1 \).

2. **Calculate the area of the regular hexagon.**
   - The formula for the area of a regular hexagon with side length \( s \) is:
     \[
     A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2
     \]
   - Substituting \( s = 1 \):
     \[
     A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \cdot 1^2 = \frac{3\sqrt{3}}{2}
     \]

3. **Determine the radius of the incircle.**
   - The radius \( r \) of the incircle of a regular hexagon with side length \( s \) is given by:
     \[
     r = \frac{s \sqrt{3}}{2}
     \]
   - Substituting \( s = 1 \):
     \[
     r = \frac{1 \cdot \sqrt{3}}{2} = \frac{\sqrt{3}}{2}
     \]

4. **Calculate the area of the incircle.**
   - The area of a circle with radius \( r \) is:
     \[
     A_{\text{incircle}} = \pi r^2
     \]
   - Substituting \( r = \frac{\sqrt{3}}{2} \):
     \[
     A_{\text{incircle}} = \pi \left( \frac{\sqrt{3}}{2} \right)^2 = \pi \cdot \frac{3}{4} = \frac{3\pi}{4}
     \]

5. **Find the ratio of the area of the incircle to the area of the hexagon.**
   - The ratio is given by:
     \[
     \text{Ratio} = \frac{A_{\text{incircle}}}{A_{\text{hexagon}}} = \frac{\frac{3\pi}{4}}{\frac{3\sqrt{3}}{2}} = \frac{3\pi}{4} \cdot \frac{2}{3\sqrt{3}} = \frac{3\pi \cdot 2}{4 \cdot 3\sqrt{3}} = \frac{6\pi}{12\sqrt{3}} = \frac{\pi}{2\sqrt{3}} = \frac{\pi \sqrt{3}}{6}
     \]

The final answer is \(\boxed{\frac{\pi \sqrt{3}}{6}}\).