What is the perimeter of a square which has the same area as a circle with circumference of 4π?
To determine the perimeter of the square, we need to find the area of the circle first. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
In this case, we are given that the circumference is 4π, so we can set up the equation as follows:
4π = 2πr
To solve for the radius, we divide both sides of the equation by 2π:
r = 4π / 2π
r = 2
Now that we have the radius, we can calculate the area of the circle using the formula A = πr^2, where A is the area and r is the radius:
A = π(2)^2
A = 4π
Since the circle has an area of 4π, the square with the same area will have a side length equal to the square root of 4π.
Side length = √(4π)
Next, we can find the perimeter of the square by multiplying the side length by 4:
Perimeter = 4 * Side length
Perimeter = 4 * √(4π)
Simplifying further, we can simplify the square root of 4 to 2:
Perimeter = 4 * 2 * √π
Perimeter = 8√π
Therefore, the perimeter of the square that has the same area as a circle with a circumference of 4π is 8√π.
To find the perimeter of a square with the same area as a circle with a circumference of 4π, we first need to find the radius of the circle.
The formula to calculate the circumference of a circle is C = 2πr, where C represents the circumference and r represents the radius.
Since the given circumference of the circle is 4π, we can substitute this value into the formula and solve for the radius:
4π = 2πr
Dividing both sides of the equation by 2π, we get:
r = 2
Now we have the radius of the circle, which is 2. Since the area of a circle is given by the formula A = πr², we can calculate the area of the circle:
A = π(2)²
Simplifying, we have:
A = 4π
The area of the square will be the same as the area of the circle, which is 4π.
To find the side length of the square (since all sides of a square are equal), we take the square root of the area:
Side length = √(4π)
Substituting the value of the area, we get:
Side length = √(4π) = 2√π
Finally, to find the perimeter of the square, we multiply the side length by 4 (since a square has four equal sides):
Perimeter = 4 * 2√π = 8√π
Therefore, the perimeter of the square is 8√π.
A = pi * r^2
A = 3.14 * 2^2
A = 12.56
√12.56 = 3.544
3.544 * 4 = perimeter
c = 2πr = 4π
so, r=2
a = πr^2 = 4π
A square with area 4π has side √(4π) = 2√π
So, its perimeter is 4*2√π = 8√π
Note that 8√π > 4π. A circle has minimum perimeter for a given area. Or, a circle has maximum area for a given perimeter.