Calculate Rectangle Area: Similar Shape with Side Length 6

To solve the problem of finding the area of a similar polygon, we first need to establish the area of the original polygon, which is a rectangle.

Step 1: Calculate the Area of the Original Rectangle
The rectangle's sides are given as $AB = 3$ and $AD = 5$. The area of the rectangle $ABCD$ is calculated as:
$\text{Area of } ABCD = AB \times AD = 3 \times 5 = 15$.

Step 2: Determine the Side Ratio
For similar polygons, the sides are proportional. We know $A'B' = 6$ is the corresponding side to $AB = 3$. Therefore, the ratio of side lengths is:
$\frac{A'B'}{AB} = \frac{6}{3} = 2$.

Step 3: Apply the Area Ratio Formula
The ratio of the areas of similar polygons is the square of the side length ratio. Therefore, the area of the new polygon can be calculated as:
$\text{Area of } A'B'C'D' = \left(\frac{\text{Side ratio}}{\text{Original area}}\right)^2 = \left(2\right)^2 \times 15 = 4 \times 15 = 60$.

Therefore, the area of the similar polygon is $\boxed{60}$.