1. We are given two similar shapes, R and S. 2. Shape R has a perimeter of $180$ mm and area of $1692$ mm$^2$. 3. Shape S has a perimeter of $300$ mm. We need to find the area of shape S. 4. Since the shapes are similar, the ratio of their perimeters is equal to the scale factor, and the ratio of their areas is the square of the scale factor. 5. Let the scale factor from shape R to shape S be $k$. $$k = \frac{\text{Perimeter of S}}{\text{Perimeter of R}} = \frac{300}{180} = \frac{5}{3}$$ 6. The area scale factor is $k^2 = \left(\frac{5}{3}\right)^2 = \frac{25}{9}$. 7. The area of shape S is then: $$\text{Area of S} = \text{Area of R} \times k^2 = 1692 \times \frac{25}{9}$$ 8. Calculate the area: $$1692 \times \frac{25}{9} = 1692 \times 2\frac{7}{9} = 1692 \times 2.777\approx 4693.3$$ 9. Rounded to one decimal place, the area of shape S is $4693.3$ mm$^2$. **Final answer:** The area of shape S is $4693.3$ mm$^2$.