S-Corporation designs its logo by linking together $4$ semicircles along the diameter of a unit circle. Find the perimeter of the shaded portion of the logo.
[img]https://cdn.artofproblemsolving.com/attachments/8/6/f0eabd46f5f3a5806d49012b2f871a453b9e7f.png[/img]
1. **Identify the radii of the semicircles:**
   - The unit circle has a radius of 1.
   - The semicircles are linked along the diameter of the unit circle.
   - Let the radius of the smaller semicircle be \( r \).
   - The radius of the larger semicircle will then be \( 1 - r \).

2. **Calculate the perimeter of each semicircle:**
   - The perimeter of a semicircle is given by \( \pi \times \text{radius} \).
   - For the smaller semicircle with radius \( r \), the perimeter is \( \pi r \).
   - For the larger semicircle with radius \( 1 - r \), the perimeter is \( \pi (1 - r) \).

3. **Sum the perimeters of all semicircles:**
   - There are two smaller semicircles and two larger semicircles.
   - The total perimeter of the smaller semicircles is \( 2 \pi r \).
   - The total perimeter of the larger semicircles is \( 2 \pi (1 - r) \).

4. **Combine the perimeters:**
   \[
   \text{Total perimeter} = 2 \pi r + 2 \pi (1 - r)
   \]
   Simplify the expression:
   \[
   \text{Total perimeter} = 2 \pi r + 2 \pi - 2 \pi r = 2 \pi
   \]

5. **Add the perimeter of the unit circle:**
   - The perimeter of the unit circle is \( 2 \pi \times 1 = 2 \pi \).

6. **Sum the perimeters of the unit circle and the semicircles:**
   \[
   \text{Total perimeter} = 2 \pi + 2 \pi = 4 \pi
   \]

The final answer is \(\boxed{4\pi}\)