The area $A$ of a circle is given by the formula:

$A_{\text{circle}} = \pi r^2$

For a circle with radius $r = 2$:

$A_{\text{circle}} = \pi (2^2) = 4\pi$

Now, for a square, the area is:

$A_{\text{square}} = s^2$

where $s$ is the side length of the square. Since the areas are equal, we set the two areas equal to each other:

$s^2 = 4\pi$

To find $s$, take the square root of both sides:

$s = \sqrt{4\pi} = 2\sqrt{\pi}$

So, the side length of the square is $2\sqrt{\pi}$, which is approximately:

$s \approx 3.5449$

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How do we derive the formula for the area of a circle?
2. What is the formula for the circumference of a circle?
3. If the radius of the circle were doubled, how would that affect the side length of the square?
4. How can we compare the perimeters of a square and a circle with equal areas?
5. What is the relation between the area and the diagonal of a square?

Tip: When comparing areas of different shapes, always remember to use the appropriate formula for each shape and set their areas equal if they are said to have the same area!