A rectangular sheet of paper whose dimensions are $12 \times  18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]
1. **Understanding the Problem:**
   We have a rectangular sheet of paper with dimensions \(12 \times 18\). When folded along a diagonal, it creates an \(M\)-shaped region. We need to find the area of the shaded region.

2. **Identify the Diagonal:**
   The diagonal of the rectangle can be calculated using the Pythagorean theorem:
   \[
   \text{Diagonal} = \sqrt{12^2 + 18^2} = \sqrt{144 + 324} = \sqrt{468} = 6\sqrt{13}
   \]

3. **Area of the Rectangle:**
   The area of the original rectangle is:
   \[
   \text{Area of Rectangle} = 12 \times 18 = 216
   \]

4. **Understanding the Fold:**
   When the paper is folded along the diagonal, it creates two congruent right triangles. Each triangle has an area of:
   \[
   \text{Area of Each Triangle} = \frac{1}{2} \times 12 \times 18 = 108
   \]

5. **Analyzing the \(M\)-shaped Region:**
   The \(M\)-shaped region consists of two parts:
   - The flat isosceles triangle \(T\) formed by the overlap.
   - The two regions outside \(T\) but inside the \(M\)-shape.

6. **Area of Triangle \(T\):**
   The triangle \(T\) is formed by the overlap of the two right triangles. To find the area of \(T\), we need to understand the ratio of the sides. The ratio between the sides of the right triangle formed by dividing \(T\) is:
   \[
   \frac{3\sqrt{13}}{18}
   \]
   This ratio simplifies to:
   \[
   \frac{\sqrt{13}}{6}
   \]

7. **Calculating the Area of \(T\):**
   The area of \(T\) can be calculated as:
   \[
   \text{Area of } T = 2 \times \left(\frac{13}{36} \times 108\right) = 2 \times 39 = 78
   \]

8. **Area of the Regions Outside \(T\):**
   Each of the two regions outside \(T\) but inside the \(M\)-shape has an area of:
   \[
   \text{Area of Each Region} = 108 - 78 = 30
   \]

9. **Total Area of the \(M\)-shaped Region:**
   The total area of the \(M\)-shaped region is the sum of the area of \(T\) and twice the area of the regions outside \(T\):
   \[
   \text{Total Area} = 78 + 2 \times 30 = 78 + 60 = 138
   \]

The final answer is \(\boxed{138}\).