To solve the problem, let’s analyze the details from the image and the information provided:

1. Square side length: The side length of the square is given as $\frac{2}{\sqrt{\pi}}$.

2. Circle inside the square: The circle fits perfectly inside the square, meaning the diameter of the circle equals the side length of the square.

3. Tasks to calculate:

   • The area of the circle.
   • The perimeter (circumference) of the circle.
   • The area of the square.
   • The perimeter of the square.

Step-by-step solution:

1. Side length of the square:

The side length of the square is: $s = \frac{2}{\sqrt{\pi}}$

2. Diameter and radius of the circle:

Since the circle fits perfectly inside the square: $\text{Diameter of the circle} = s = \frac{2}{\sqrt{\pi}}$ $\text{Radius of the circle} = r = \frac{\text{Diameter}}{2} = \frac{1}{\sqrt{\pi}}$

3. Area of the circle:

The formula for the area of a circle is: $\text{Area of the circle} = \pi r^2$ Substituting $r = \frac{1}{\sqrt{\pi}}$: $\text{Area of the circle} = \pi \left(\frac{1}{\sqrt{\pi}}\right)^2 = \pi \cdot \frac{1}{\pi} = 1$

4. Perimeter (Circumference) of the circle:

The formula for the perimeter (circumference) of a circle is: $\text{Perimeter of the circle} = 2\pi r$ Substituting $r = \frac{1}{\sqrt{\pi}}$: $\text{Perimeter of the circle} = 2\pi \cdot \frac{1}{\sqrt{\pi}} = 2\sqrt{\pi}$

5. Area of the square:

The formula for the area of a square is: $\text{Area of the square} = s^2$ Substituting $s = \frac{2}{\sqrt{\pi}}$: $\text{Area of the square} = \left(\frac{2}{\sqrt{\pi}}\right)^2 = \frac{4}{\pi}$

6. Perimeter of the square:

The formula for the perimeter of a square is: $\text{Perimeter of the square} = 4s$ Substituting $s = \frac{2}{\sqrt{\pi}}$: $\text{Perimeter of the square} = 4 \cdot \frac{2}{\sqrt{\pi}} = \frac{8}{\sqrt{\pi}}$

Final Results:

1. Area of the circle: $1$ (square units).
2. Perimeter of the circle: $2\sqrt{\pi}$ (units).
3. Area of the square: $\frac{4}{\pi}$ (square units).
4. Perimeter of the square: $\frac{8}{\sqrt{\pi}}$ (units).

Would you like a deeper explanation or further clarification?
Here are 5 related questions you might explore:

1. How does the square's area compare to the circle's area?
2. What is the ratio of the square's perimeter to the circle's circumference?
3. How would these calculations change if the square's side length were doubled?
4. Can you generalize these formulas for a square with side $s$?
5. What is the percentage of the square's area that is covered by the circle?

Tip: For geometric problems, always look for symmetries (like the circle's diameter equaling the square's side) to simplify calculations.