To solve this problem, let's work step by step:

Given:

1. Shape C:
   • Perimeter = $180 \, \text{mm}$
   • Area = $1332 \, \text{mm}^2$
2. Shape D:
   • Perimeter = $300 \, \text{mm}$

We need to calculate the area of shape D, and both shapes are similar.

Step 1: Calculate the scale factor between the perimeters

The scale factor $k$ for similar shapes is given by the ratio of their corresponding linear dimensions, such as the perimeters:

$k = \frac{\text{Perimeter of Shape D}}{\text{Perimeter of Shape C}} = \frac{300}{180} = \frac{5}{3}$

So, the scale factor $k$ is $\frac{5}{3}$.

Step 2: Relating areas for similar shapes

For similar shapes, the ratio of the areas is the square of the scale factor. That is:

$\frac{\text{Area of Shape D}}{\text{Area of Shape C}} = k^2$

Substitute $k = \frac{5}{3}$:

$\frac{\text{Area of Shape D}}{\text{1332}} = \left(\frac{5}{3}\right)^2$

$\frac{\text{Area of Shape D}}{1332} = \frac{25}{9}$

Step 3: Solve for the area of Shape D

$\text{Area of Shape D} = 1332 \times \frac{25}{9}$

$\text{Area of Shape D} = \frac{1332 \times 25}{9} = \frac{33300}{9} \approx 3700.0 \, \text{mm}^2$

Thus, the area of Shape D is approximately 3700.0 mm².

Would you like further clarification or detailed steps?
Here are some related questions to deepen understanding:

1. How do scale factors affect the relationship between perimeters, areas, and volumes?
2. If a third shape is introduced with a different perimeter, how can we compare its area to Shape C and D?
3. How does this problem change if the shapes are not similar?
4. Can you use the same process for 3D shapes, such as calculating surface areas and volumes?
5. How would rounding affect precision in the final result?

Tip: Always double-check your calculations when working with scale factors, especially when squaring or cubing them!