A strip of paper 60 cm long is folded to form the shape shown below. The difference in area between the strip of paper and the folded strip is 18 cm$^2$. Find the area of the shaded folded strip. Ans: ___ $\text{cm}^2$

1. Understand the relationship between the original strip area, the folded shape area, and the overlapping regions.

When the strip is folded into a rectangular frame, there are 4 corners where the paper overlaps itself. Each corner forms an isosceles right-angled triangle of double thickness. The difference in area between the unfolded strip and the folded shape is exactly the area of these 4 overlapping triangles (since they are counted twice in the unfolded strip but only once in the visible folded shape).

2. Determine the area of the 4 overlapping triangles.

Difference in area = Area of 4 overlapping triangles = $18 \text{ cm}^2$.

3. Find the width of the strip.

Let the width of the strip be $w$. Each overlapping triangle is an isosceles right-angled triangle with legs equal to $w$. Area of 1 triangle = $\frac{1}{2} w^2$.
Area of 4 triangles = $4 \times \frac{1}{2} w^2 = 2w^2$.
$2w^2 = 18 \implies w^2 = 9 \implies w = 3 \text{ cm}$.

4. Calculate the area of the original strip.

Area of original strip = Length $\times$ Width = $60 \times 3 = 180 \text{ cm}^2$.

5. Calculate the area of the shaded folded strip.

Area of folded shape = Area of original strip - Difference in area
Area = $180 - 18 = 162 \text{ cm}^2$.