Which of the following is the circumference of the circle whose area is equal to the area of a square whose perimeter is 49 inches?
The correct answer and explanation is :
To find the circumference of the circle whose area is equal to the area of a square with a given perimeter, we need to break down the problem step by step.
Step 1: Determine the side length of the square.
We are told that the perimeter of the square is 49 inches. The perimeter ( P ) of a square is calculated as:
[
P = 4 \times \text{side length}
]
Given ( P = 49 ) inches, we can solve for the side length of the square:
[
49 = 4 \times \text{side length}
]
[
\text{side length} = \frac{49}{4} = 12.25 \text{ inches}
]
Step 2: Calculate the area of the square.
The area ( A_{\text{square}} ) of a square is given by:
[
A_{\text{square}} = \text{side length}^2
]
Substituting the side length we found:
[
A_{\text{square}} = 12.25^2 = 150.0625 \text{ square inches}
]
Step 3: Find the radius of the circle with the same area.
Next, we need to find the radius of a circle whose area is equal to the area of the square. The area ( A_{\text{circle}} ) of a circle is given by:
[
A_{\text{circle}} = \pi r^2
]
where ( r ) is the radius of the circle. Since the area of the circle is equal to the area of the square, we set the two areas equal:
[
\pi r^2 = 150.0625
]
Now, solve for ( r^2 ):
[
r^2 = \frac{150.0625}{\pi}
]
[
r^2 = \frac{150.0625}{3.1416} \approx 47.75
]
[
r = \sqrt{47.75} \approx 6.91 \text{ inches}
]
Step 4: Calculate the circumference of the circle.
The circumference ( C ) of a circle is given by the formula:
[
C = 2\pi r
]
Substituting the radius we found:
[
C = 2 \times 3.1416 \times 6.91 \approx 43.42 \text{ inches}
]
Thus, the circumference of the circle is approximately 43.42 inches.
Conclusion:
The circle whose area is equal to the area of the square with a perimeter of 49 inches has a circumference of approximately 43.42 inches.