a wire in the shape of circle encloses the area 38.5 cm square if the wire is rebent to form a square find the area of the square
To find the area of the square formed by rebending the wire, we need to determine the length of each side of the square.
The wire is in the shape of a circle, and we know that the area enclosed by the wire is 38.5 cm². The formula for the area of a circle is given by A = πr², where A is the area and r is the radius.
Given that the wire is in the shape of a circle, the perimeter of the wire (which will be the same as the length of the square) is given by the circumference of the circle. The formula for the circumference is C = 2πr.
We can find the radius (r) by rearranging the formula for the area of the circle:
A = πr²
38.5 = πr²
r² = 38.5/π
r = √(38.5/π)
To find the length of each side of the square, we need to determine the circumference of the circle by using the radius:
C = 2πr
C = 2π√(38.5/π)
Now, we know the length of each side of the square (s) is equal to the circumference of the circle:
s = 2π√(38.5/π)
Finally, we can calculate the area of the square (A) by squaring the length of each side:
A = s²
A = (2π√(38.5/π))²
You can simplify this further to get the final result.
To start, let's calculate the length of the wire in the shape of a circle. We know that the circumference of a circle is given by the formula:
C = 2πr
where C is the circumference and r is the radius of the circle. Since we don't have the radius, we need to find it using the area.
The formula for the area of a circle is:
A = πr^2
Given that the area is 38.5 cm^2, we can solve for r:
38.5 = πr^2
Dividing both sides by π, we get:
38.5 / π = r^2
To isolate r, we take the square root of both sides:
√(38.5 / π) = r
Now that we have the radius, we can calculate the circumference of the circle:
C = 2πr
C = 2π * √(38.5 / π)
Simplifying, we get:
C = 2√(38.5π)
Next, we can determine the length of each side of the square. Since the wire is rebent to form a square, the length of the wire will be equal to the perimeter of the square, which is four times the length of one side:
Perimeter of Square = 4 * Side Length
Since the perimeter of the square is equal to the length of the wire, we can set up the following equation:
4 * Side Length = C
Now, we can substitute the value of C we found earlier:
4 * Side Length = 2√(38.5π)
Dividing both sides by 4:
Side Length = √(38.5π) / 2
Finally, we can calculate the area of the square, knowing that the area of a square is given by:
A = Side Length * Side Length
Plugging in the value of Side Length:
A = (√(38.5π) / 2) * (√(38.5π) / 2)
Simplifying, we get:
A = (38.5π) / 4
Calculating the area using the value of π (approximately 3.14159):
A ≈ (38.5 * 3.14159) / 4
A ≈ 30.37118 cm^2
Therefore, the area of the square formed by rebending the wire is approximately 30.37118 cm^2.
a = pi r^2 = 38.5
r^2 = 38.5/pi
r = 3.50
c = 2pi r = 22.06
Now, for the square:
s = 22.06/4 = 5.515
a = 30.42