The question asks us to calculate the area of shape S, given that shapes R and S are similar. We have the following information:

• The perimeter of shape R is 180 mm, and its area is 1692 mm².
• The perimeter of shape S is 300 mm.

Since the shapes are similar, their corresponding linear dimensions are proportional. The ratio of their perimeters will be the same as the ratio of their corresponding side lengths.

Step 1: Find the ratio of the perimeters

$\text{Ratio of perimeters} = \frac{\text{Perimeter of shape S}}{\text{Perimeter of shape R}} = \frac{300}{180} = \frac{5}{3}$

Step 2: Relationship between the areas

Since the shapes are similar, the ratio of their areas will be the square of the ratio of their corresponding side lengths (or perimeters).

$\text{Ratio of areas} = \left(\frac{5}{3}\right)^2 = \frac{25}{9}$

Step 3: Calculate the area of shape S

We know the area of shape R is 1692 mm², so the area of shape S will be:

$\text{Area of shape S} = \text{Area of shape R} \times \frac{25}{9} = 1692 \times \frac{25}{9} = 4700 \text{ mm}^2$

Thus, the area of shape S is 4700 mm².

Let me know if you'd like more details or if you have any questions!

Here are some related questions:

1. How do we use similarity ratios to solve other types of geometric problems?
2. How does the ratio of areas change if we change the ratio of perimeters?
3. What are other real-world applications of shape similarity in mathematics?
4. How would the problem change if the ratio of the sides was reversed?
5. Can we extend this method to 3D shapes?

Tip: When working with similar shapes, remember that the ratio of areas is the square of the ratio of their sides or perimeters!