To find the area of a circle whose circumference equals the perimeter of a square, we'll first calculate the perimeter of the square, then the circumference of the circle, and finally the area of the circle.
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Calculate the perimeter of the square:
- The formula for the perimeter \( P \) of a square is given by: \[ P = 4 \times \text{side length} \]
- Given the side length is 5.5 units: \[ P = 4 \times 5.5 = 22 \text{ units} \]
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Set the circumference of the circle equal to the perimeter of the square:
- The formula for the circumference \( C \) of a circle is given by: \[ C = 2\pi r \]
- Since we want the circumference to equal 22 units: \[ 2\pi r = 22 \]
- We can solve for \( r \): \[ r = \frac{22}{2\pi} = \frac{22}{2 \times 3.14} = \frac{22}{6.28} \approx 3.5 \text{ units} \]
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Calculate the area of the circle:
- The formula for the area \( A \) of a circle is: \[ A = \pi r^2 \]
- Substituting \( r = 3.5 \) units: \[ A = 3.14 \times (3.5)^2 = 3.14 \times 12.25 \approx 38.47 \text{ units}^2 \]
Thus, the area of the circle is approximately 38.47 units².
The answer is:
38.47 units²
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