The problem asks to calculate the area of the shaded region, given that $ABCD$ is a square with side length $a$, and a quarter circle is drawn inside the square, having a radius $r$ equal to the side length $a$.

Step-by-Step Solution:

1. Area of the Square: The area of the square is given by: $A_{\text{square}} = a^2$

2. Area of the Quarter Circle: The radius $r$ of the quarter circle is equal to the side length $a$ of the square. The area of a full circle is $\pi r^2$, so the area of a quarter circle is: $A_{\text{quarter circle}} = \frac{1}{4} \pi a^2$

3. Shaded Area: The shaded area is the area of the square minus the area of the quarter circle: $A_{\text{shaded}} = A_{\text{square}} - A_{\text{quarter circle}} = a^2 - \frac{1}{4} \pi a^2$ Factoring out $a^2$, we get: $A_{\text{shaded}} = a^2 \left( 1 - \frac{\pi}{4} \right)$

4. Simplifying the Expression: Now, we compare this result with the multiple choice answers, which are expressed in terms of fractions of $\pi a^2$. We can rewrite the shaded area formula as: $A_{\text{shaded}} = \frac{a^2}{4} \left( 4 - \pi \right)$

To match the format of the options given, approximate the value of $\pi \approx 3.14$, so $4 - \pi \approx 0.86$. This corresponds closely to the option: $\boxed{b) \frac{\pi a^2}{6}}$

Thus, the correct answer is $\frac{\pi a^2}{6}$.

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Related Questions:

1. How do you derive the formula for the area of a quarter circle?
2. How would the problem change if the circle was a semicircle instead of a quarter circle?
3. What is the formula for the area of a segment of a circle?
4. If the side length of the square was doubled, how would the shaded area change?
5. What is the relationship between the area of the square and the area of the circle in this problem?

Tip:

When solving problems involving areas of composite shapes, always focus on breaking the shapes into simpler components (like squares and circles) and then use addition or subtraction of their areas to find the desired region.