a square in a circle with a radius of 2 pi .What is the perimeter of the square?
radius = 2pi, so diameter = 4pi
circle diameter is the length of the side of the square.
perimeter = 4*4pi = 16pi
Isn't the diagonal of the square the same as the diameter of the circle?
http://flowers4u.files.wordpress.com/2009/10/square-within-circle-fig-10-gt.jpg
So the diagonal of the square is 4π
let each side of square be x
x^2 + x^2 = 16π^2
2x^2 = 16π^2
x^2 = 8π^2
x = π√8
4x = 4π√8
= 8√2 π
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To find the perimeter of the square, we first need to determine the length of one side of the square.
Given that the circle has a radius of 2π, the diameter of the circle is 2 times the radius, which is 2π * 2 = 4π. Since the diameter of the circle is also equal to the diagonal of the square, we can use the diagonal of the square to find the side length.
In a square, the diagonal divides the square into two congruent right triangles. Each right triangle has a hypotenuse equal to the length of the diagonal and the base and height equal to the side length of the square.
Applying the Pythagorean theorem, we can solve for the side length of the square:
(diagonal)^2 = (base)^2 + (height)^2
(4π)^2 = (side)^2 + (side)^2
16π^2 = 2(side)^2
8π^2 = (side)^2
Taking the square root of both sides, we find:
√(8π^2) = side
√8π^2 = side
2√2π = side
Now that we have the length of one side of the square, we can find the perimeter by multiplying the side length by 4 (since all four sides of a square are equal in length):
Perimeter = 4 * side
Perimeter = 4 * 2√2π
Perimeter = 8√2π
Therefore, the perimeter of the square is 8√2π.