Let $ABCDEF$ be a regular hexagon with side length $2$. Calculate the area of $ABDE$.

[i]2015 CCA Math Bonanza Lightning Round #1.2[/i]
1. **Identify the structure of the regular hexagon:**
   A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is equal to the side length of these triangles, which is given as 2.

2. **Determine the coordinates of the vertices:**
   Place the hexagon in the coordinate plane with the center at the origin. The vertices of the hexagon can be represented as:
   \[
   A = (2, 0), \quad B = (1, \sqrt{3}), \quad C = (-1, \sqrt{3}), \quad D = (-2, 0), \quad E = (-1, -\sqrt{3}), \quad F = (1, -\sqrt{3})
   \]

3. **Identify the vertices of quadrilateral \(ABDE\):**
   The vertices of \(ABDE\) are \(A = (2, 0)\), \(B = (1, \sqrt{3})\), \(D = (-2, 0)\), and \(E = (-1, -\sqrt{3})\).

4. **Calculate the lengths of the sides of \(ABDE\):**
   - The length of \(AB\) is the side length of the hexagon, which is 2.
   - The length of \(AD\) can be calculated using the distance formula:
     \[
     AD = \sqrt{(2 - (-2))^2 + (0 - 0)^2} = \sqrt{4^2} = 4
     \]
   - The length of \(DE\) is the side length of the hexagon, which is 2.
   - The length of \(BE\) can be calculated using the distance formula:
     \[
     BE = \sqrt{(1 - (-1))^2 + (\sqrt{3} - (-\sqrt{3}))^2} = \sqrt{2^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4
     \]

5. **Calculate the area of \(ABDE\):**
   Since \(ABDE\) is a parallelogram (as opposite sides are equal and parallel), we can use the formula for the area of a parallelogram:
   \[
   \text{Area} = \text{base} \times \text{height}
   \]
   Here, the base is \(AB = 2\) and the height is the perpendicular distance between \(AB\) and \(DE\), which is the same as the height of the equilateral triangle formed by the side length of the hexagon:
   \[
   \text{height} = \sqrt{3} \times \text{side length} = \sqrt{3} \times 2 = 2\sqrt{3}
   \]
   Therefore, the area of \(ABDE\) is:
   \[
   \text{Area} = 2 \times 2\sqrt{3} = 4\sqrt{3}
   \]

The final answer is \(\boxed{4\sqrt{3}}\).