Find the area of the figure  surrounded by the locus of the point $ P$ inside the square with side length $ a$ such that the distance to the center of the square is equal to the minimum distance to the side of the square.
1. **Place the square in the coordinate system:**
   - Consider a square centered at the origin with side length \(a\). The vertices of the square are at \((\pm \frac{a}{2}, \pm \frac{a}{2})\).

2. **Define the point \(P(x, y)\):**
   - Let \(P(x, y)\) be a point inside the square such that the distance from \(P\) to the center of the square is equal to the minimum distance from \(P\) to any side of the square.

3. **Distance to the center:**
   - The distance from \(P(x, y)\) to the center \((0, 0)\) is given by:
     \[
     d_{\text{center}} = \sqrt{x^2 + y^2}
     \]

4. **Minimum distance to the side:**
   - The minimum distance from \(P(x, y)\) to any side of the square is:
     \[
     d_{\text{side}} = \min\left(\frac{a}{2} - |x|, \frac{a}{2} - |y|\right)
     \]

5. **Equating the distances:**
   - According to the problem, these distances are equal:
     \[
     \sqrt{x^2 + y^2} = \min\left(\frac{a}{2} - |x|, \frac{a}{2} - |y|\right)
     \]

6. **Analyzing the symmetry:**
   - Due to the symmetry of the problem, we can consider only the first quadrant where \(x \geq 0\) and \(y \geq 0\). The equation simplifies to:
     \[
     \sqrt{x^2 + y^2} = \frac{a}{2} - x \quad \text{or} \quad \sqrt{x^2 + y^2} = \frac{a}{2} - y
     \]

7. **Solving for the curve:**
   - Consider the case \(\sqrt{x^2 + y^2} = \frac{a}{2} - x\):
     \[
     x^2 + y^2 = \left(\frac{a}{2} - x\right)^2
     \]
     \[
     x^2 + y^2 = \frac{a^2}{4} - ax + x^2
     \]
     \[
     y^2 = \frac{a^2}{4} - ax
     \]
     \[
     y = \sqrt{\frac{a^2}{4} - ax}
     \]

8. **Finding the area of the parabolic section:**
   - The height of the parabolic section is \(\frac{a}{8}\) and the width is \(\frac{a}{2}\). The area of one parabolic section is:
     \[
     \text{Area} = \frac{2}{3} \cdot \frac{a}{2} \cdot \frac{a}{8} = \frac{a^2}{24}
     \]

9. **Total area of the parabolic sections:**
   - There are four such parabolic sections, so the total area is:
     \[
     4 \cdot \frac{a^2}{24} = \frac{a^2}{6}
     \]

10. **Area of the central square:**
    - The central square has side length \(\frac{a}{2}\), so its area is:
      \[
      \left(\frac{a}{2}\right)^2 = \frac{a^2}{4}
      \]

11. **Total area:**
    - The total area of the figure is the sum of the areas of the parabolic sections and the central square:
      \[
      \frac{a^2}{6} + \frac{a^2}{4} = \frac{2a^2}{12} + \frac{3a^2}{12} = \frac{5a^2}{12}
      \]

The final answer is \(\boxed{\frac{5a^2}{12}}\)