what is the length of one side of a square that has the same area as a circle with raidus 2

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!

What is the length of one side of a square that has the same area as a circle with radius 2?

Discover how to calculate the side length of a square that has the same area as a circle with a radius of 2. This geometry problem involves understanding the formulas for the area of a circle and square, and applying square root concepts. Ideal for students in Grades 8-10.

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