To solve this problem, let's follow these steps:
1. Find the perimeter of the square:
Since a square has four equal sides, the perimeter (P) is calculated by adding the lengths of all four sides. If one side of the square is 5.5 units, then the perimeter is:
[tex]\( P_{square} = 4 \times \text{side of the square} = 4 \times 5.5 = 22 \text{ units} \)[/tex]
2. Use the perimeter of the square as the circumference of the circle:
The problem states that the circumference (C) of the circle is equal to the perimeter of the square. Thus:
[tex]\( C_{circle} = P_{square} = 22 \text{ units} \)[/tex]
3. Find the radius of the circle using the circumference:
The formula for the circumference of a circle is [tex]\( C = 2 \pi r \)[/tex], where [tex]\( \pi \approx 3.14 \)[/tex] and [tex]\( r \)[/tex] is the radius of the circle. We can rearrange the formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\( r = \frac{C}{2 \pi} \)[/tex]
Plugging in the values:
[tex]\( r = \frac{22}{2 \times 3.14} = \frac{22}{6.28} \approx 3.5 \text{ units} \)[/tex]
4. Calculate the area of the circle:
The formula for the area (A) of a circle is [tex]\( A = \pi r^2 \)[/tex]. Using the radius we just found:
[tex]\( A = 3.14 \times (3.5)^2 \approx 3.14 \times 12.25 \)[/tex]
[tex]\( A \approx 3.14 \times 12.25 \approx 38.465 \text{ units}^2 \)[/tex]
5. Round the area to the nearest hundredth:
[tex]\( A \approx 38.47 \text{ units}^2 \)[/tex]
Therefore, the area of the circle is approximately 38.47 square units, which matches one of the multiple-choice options provided.
