The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png[/img]
1. **Understanding the Problem:**
   The problem involves a $12$ by $20$ rectangle split into four strips of equal widths, surrounding an isosceles triangle. We need to find the area of the shaded region.

2. **Identifying the Shapes:**
   The rectangle is divided into:
   - Two right triangles
   - One isosceles triangle

3. **Calculating the Area of the Right Triangles:**
   Each right triangle has a base of $10$ (half of $20$) and a height of $12$.
   \[
   \text{Area of one right triangle} = \frac{1}{2} \times 10 \times 12 = 60
   \]

4. **Calculating the Shaded Area in One Right Triangle:**
   The shaded area in one right triangle is $\frac{5}{8}$ of the entire triangle.
   \[
   \text{Shaded area in one right triangle} = \frac{5}{8} \times 60 = \frac{300}{8} = 37.5
   \]

5. **Calculating the Area of the Isosceles Triangle:**
   The isosceles triangle has a base of $20$ and a height of $12$.
   \[
   \text{Area of the isosceles triangle} = \frac{1}{2} \times 20 \times 12 = 120
   \]

6. **Calculating the Shaded Area in the Isosceles Triangle:**
   The shaded area in the isosceles triangle is also $\frac{5}{8}$ of the entire triangle.
   \[
   \text{Shaded area in the isosceles triangle} = \frac{5}{8} \times 120 = \frac{600}{8} = 75
   \]

7. **Summing Up the Shaded Areas:**
   There are two right triangles, so the total shaded area in the right triangles is:
   \[
   2 \times 37.5 = 75
   \]
   Adding the shaded area of the isosceles triangle:
   \[
   75 + 75 = 150
   \]

Conclusion:
\[
\boxed{150}
\]