Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$.

[i]Proposed by Michael Tang[/i]
1. **Identify the Geometry and Setup:**
   - We have three congruent circles of radius \(2\) such that each circle passes through the centers of the other two circles.
   - The centers of these circles form an equilateral triangle \(ABC\) with side length equal to the diameter of the circles, which is \(4\).

2. **Determine the Midpoints of the Arcs:**
   - Let \(D\), \(E\), and \(F\) be the midpoints of the arcs \(AB\), \(BC\), and \(CA\) respectively.

3. **Equilateral Triangle \(ABC\):**
   - Since \(ABC\) is an equilateral triangle with side length \(4\), the height \(h\) of this triangle is given by:
     \[
     h = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3}
     \]

4. **Equilateral Triangle \(DEF\):**
   - The triangle \(DEF\) is also equilateral because the midpoints of the arcs are symmetrically placed.
   - The side length of \(DEF\) is the same as the radius of the circles, which is \(2\).

5. **Calculate the Area of Triangle \(DEF\):**
   - The area \(K\) of an equilateral triangle with side length \(s\) is given by:
     \[
     K = \frac{\sqrt{3}}{4} s^2
     \]
   - For triangle \(DEF\) with side length \(2\):
     \[
     K = \frac{\sqrt{3}}{4} \times 2^2 = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3}
     \]

6. **Express \(K\) in the Given Form:**
   - We are given that \(K = \sqrt{a} - b\). From our calculation, \(K = \sqrt{3}\).
   - Therefore, we can write:
     \[
     \sqrt{3} = \sqrt{3} - 0
     \]
   - This implies \(a = 3\) and \(b = 0\).

7. **Calculate \(100a + b\):**
   - Substitute \(a = 3\) and \(b = 0\):
     \[
     100a + b = 100 \times 3 + 0 = 300
     \]

The final answer is \(\boxed{300}\)