The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
[img]https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png[/img]
1. **Calculate the perimeter of the original $8 \times 12$ rectangle:**
   The perimeter \( P \) of a rectangle is given by the formula:
   \[
   P = 2 \times (\text{length} + \text{width})
   \]
   Here, the length is 12 units and the width is 8 units. Therefore,
   \[
   P = 2 \times (12 + 8) = 2 \times 20 = 40 \text{ units}
   \]

2. **Account for the removal of six $2 \times 2$ squares:**
   Each $2 \times 2$ square removed from the sides of the rectangle adds additional sides to the perimeter. Each square adds 4 sides, but since they are removed from the rectangle, they effectively add 2 sides each to the perimeter (the two sides that are now exposed).

3. **Calculate the additional perimeter due to the removed squares:**
   Since there are 6 squares removed, and each square adds 2 sides to the perimeter, the total additional perimeter is:
   \[
   6 \times 2 = 12 \text{ units}
   \]

4. **Combine the original perimeter with the additional perimeter:**
   The total perimeter of the polygon is the sum of the original perimeter and the additional perimeter:
   \[
   40 + 12 = 52 \text{ units}
   \]

The final answer is \(\boxed{52}\) units.