The problem asks to calculate the shaded area in the image provided. The figure consists of a quarter circle with center $O$ and radius $r$, and a square inscribed within the quarter circle.

Let's break down the solution:

1. Radius of the quarter circle: The quarter circle's radius $r$ is equal to the distance $OB$ or $OA$.

2. Side of the square: Let's denote the side of the square as $s$. Since the square is inscribed, one of its vertices lies on the circle, and the square touches the circle at two other points.

3. Relationship between the square and circle: The diagonal of the square $OP$ is the same as the radius $r$ of the quarter circle. The diagonal $OP$ can be related to the side of the square $s$ using the Pythagorean theorem: $OP = s\sqrt{2}$ But since $OP = r$, we have: $r = s\sqrt{2}$ Solving for $s$: $s = \frac{r}{\sqrt{2}}$

4. Area of the quarter circle: The area of the quarter circle is: $\text{Area of quarter circle} = \frac{1}{4} \pi r^2$

5. Area of the square: The area of the square is: $\text{Area of square} = s^2 = \left(\frac{r}{\sqrt{2}}\right)^2 = \frac{r^2}{2}$

6. Shaded area: The shaded area is the difference between the area of the quarter circle and the area of the square: $\text{Shaded area} = \frac{1}{4} \pi r^2 - \frac{r^2}{2}$ Simplifying further: $\text{Shaded area} = r^2 \left(\frac{\pi}{4} - \frac{1}{2}\right)$

So, the shaded area is $r^2 \left(\frac{\pi}{4} - \frac{1}{2}\right)$.

Would you like further details on any of these steps, or do you have any questions?

Related Questions:

1. How would the shaded area change if the figure was a semicircle instead of a quarter circle?
2. What would the relationship be if the square was not inscribed but circumscribed around the quarter circle?
3. Can you derive the area of the square in terms of the circumference of the full circle?
4. What happens to the shaded area if the square is scaled down or up?
5. How would you calculate the shaded area if the square was rotated within the quarter circle?

Tip:

When dealing with geometry problems involving circles and polygons, it’s often useful to explore relationships between the radius, side lengths, and angles using trigonometry or the Pythagorean theorem.