To find the side length $x$ of a square with the same area as a circle with a radius of 10 cm, we need to set up the relationship between the areas.

Step 1: Area of the Circle

The formula for the area of a circle is:

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

Given that the radius $r = 10$ cm, the area of the circle is:

$A_{\text{circle}} = \pi (10)^2 = 100\pi \, \text{cm}^2$

Step 2: Area of the Square

The area of a square with side length $x$ is:

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

Step 3: Set the Areas Equal

Since the areas of the circle and the square are equal, we can set the two area formulas equal to each other:

$x^2 = 100\pi$

Step 4: Solve for $x$

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

$x = \sqrt{100\pi}$

Now, use the approximation $\pi \approx 3.1416$:

$x = \sqrt{100 \times 3.1416} = \sqrt{314.16} \approx 17.7 \, \text{cm}$

Final Answer:

The side length of the square is approximately $x \approx 17.7$ cm.

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Would you like further clarification or have any questions? Here are some related questions to explore:

1. How does the area of a circle relate to the radius?
2. How can you calculate the area of other shapes like triangles and rectangles?
3. What is the relationship between the area and perimeter of a square?
4. Can the method used here be applied to find side lengths for other shapes with equivalent areas?
5. What happens to the side length of the square if the radius of the circle changes?

Tip: When solving problems involving areas of different shapes, always make sure the formulas match the type of shape you're working with!