An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap).
1. **Determine the side length of the equilateral triangle and the square:**
   Since the equilateral triangle \(ABC\) shares a side with the square \(BCDE\), all sides of the triangle and the square are equal. Let the side length be \(s\).

2. **Calculate the perimeter of the pentagon:**
   The pentagon consists of 5 sides: 3 sides from the equilateral triangle \(ABC\) and 2 additional sides from the square \(BCDE\) (since side \(BC\) is shared). Therefore, the perimeter of the pentagon is:
   \[
   5s = 20
   \]
   Solving for \(s\):
   \[
   s = \frac{20}{5} = 4
   \]

3. **Calculate the area of the square \(BCDE\):**
   The area of the square is given by:
   \[
   \text{Area of square} = s^2 = 4^2 = 16
   \]

4. **Calculate the area of the equilateral triangle \(ABC\):**
   The area of an equilateral triangle with side length \(s\) is given by:
   \[
   \text{Area of triangle} = \frac{s^2 \sqrt{3}}{4}
   \]
   Substituting \(s = 4\):
   \[
   \text{Area of triangle} = \frac{4^2 \sqrt{3}}{4} = \frac{16 \sqrt{3}}{4} = 4\sqrt{3}
   \]

5. **Calculate the total area of the pentagon:**
   Since the triangle and the square do not overlap, the total area of the pentagon is the sum of the areas of the square and the triangle:
   \[
   \text{Total area} = \text{Area of square} + \text{Area of triangle} = 16 + 4\sqrt{3}
   \]

\[
\boxed{16 + 4\sqrt{3}}
\]