To solve this problem, we will find the area of the shaded portion in the square.

Step-by-step solution:

1. Analyze the geometry:

   • The figure is a square with a side length of 6.
   • The arc shown is a quarter-circle with its center at one corner of the square, and its radius equal to the side length of the square (6 units).

2. Area of the square: $\text{Area of the square} = \text{side}^2 = 6^2 = 36 \, \text{square units}.$

3. Area of the quarter-circle: The formula for the area of a circle is: $A = \pi r^2,$ where $r$ is the radius. For a quarter-circle: $\text{Area of quarter-circle} = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (6)^2 = \frac{36 \pi}{4} = 9 \pi \, \text{square units}.$

4. Area of the shaded region: The shaded region is the part of the square that is not covered by the quarter-circle. Therefore: $\text{Shaded Area} = \text{Area of the square} - \text{Area of the quarter-circle}.$ Substitute the values: $\text{Shaded Area} = 36 - 9 \pi \, \text{square units}.$

   Since $\pi$ is approximately $3.14$: $\text{Shaded Area} \approx 36 - 9(3.14) = 36 - 28.26 = 7.74 \, \text{square units}.$

Final Answer:

The area of the shaded portion is: $\boxed{36 - 9\pi \, \text{square units}} \quad \text{or approximately } \boxed{7.74 \, \text{square units}}.$

Do you want me to explain any step further or compute anything else? Here are 5 related questions you might explore:

1. How would the shaded area change if the side length of the square were doubled?
2. What fraction of the square's total area is shaded?
3. Can you generalize the formula for the shaded area for any side length $s$ of the square?
4. How would the problem change if the arc were a semicircle instead of a quarter-circle?
5. How can you calculate the perimeter of the shaded region?

Tip: Understanding the relationship between areas and proportions is key when working with composite shapes!