To solve the problem in the given image, we will analyze the geometry and compute the area of the shaded region.

Analysis:

1. The square has side length $AB = BC = 6 \, \text{cm}$, so its area is: $\text{Area of square} = AB^2 = 6^2 = 36 \, \text{cm}^2.$

2. The shaded area is the region bounded by the two quarter-circles. Each quarter-circle has a radius equal to the side of the square ($6 \, \text{cm}$).

3. Quarter-circle area: The area of a quarter-circle is given by: $\text{Quarter-circle area} = \frac{1}{4} \pi r^2.$ For a radius of $r = 6 \, \text{cm}$: $\text{Area of one quarter-circle} = \frac{1}{4} \pi (6^2) = \frac{1}{4} \pi (36) = 9\pi \, \text{cm}^2.$

4. Intersection of the two quarter-circles: The shaded area is equal to the square's area minus the area common to the two quarter-circles. This intersection forms a lens-like shape.

The calculation involves geometric subtraction:

• Add up the areas of both quarter-circles.
• Subtract the overlapping area.

Would you like a detailed computation or diagram of how this overlapping area is determined?

Meanwhile, here are five relative questions:

1. What is the formula to calculate areas of shapes bounded by circles?
2. How would the solution change if the side length of the square doubled?
3. How do you determine the overlapping region of two quarter-circles geometrically?
4. How can you use integration to compute complex areas like this?
5. What is the relation between quarter-circles and their enclosing square?

Tip:

Always remember to break down composite geometric problems into their simpler components. This simplifies both understanding and computation.