The problem states that the perimeter of a square with side length 11 cm is equal to the circumference of a circle. We need to find the area of the circle.

Step 1: Find the perimeter of the square.

The perimeter $P$ of a square is calculated by the formula: $P = 4 \times \text{side}$ Given the side length is 11 cm: $P = 4 \times 11 = 44 \text{ cm}$ Thus, the perimeter of the square is 44 cm.

Step 2: Use the perimeter to find the radius of the circle.

The problem states that the perimeter of the square is equal to the circumference of the circle. The formula for the circumference $C$ of a circle is: $C = 2 \pi r$ Equating the circumference to the perimeter of the square: $2 \pi r = 44$ Solve for $r$ (the radius of the circle): $r = \frac{44}{2\pi} = \frac{22}{\pi} \approx 7.00 \text{ cm}$

Step 3: Find the area of the circle.

The formula for the area $A$ of a circle is: $A = \pi r^2$ Substitute $r = \frac{22}{\pi} \approx 7.00 \text{ cm}$: $A = \pi \times (7)^2 = \pi \times 49 \approx 153.94 \text{ cm}^2$

Final Answer:

The area of the circle is approximately $153.94 \, \text{cm}^2$.

Would you like more details or have any questions?

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Here are 5 related questions you could explore:

1. What would be the radius of the circle if the perimeter of the square was doubled?
2. How does the relationship between the perimeter of the square and the circle's circumference change with increasing side length?
3. What is the formula for the area of the square and how does it compare to the circle's area?
4. How can you derive the formula for the circumference of a circle?
5. What would happen if the radius of the circle was halved?

Tip: Always keep track of the units while solving geometry problems to avoid confusion with dimensions like perimeter, area, and volume.