Perhaps you are an artist or interior designer or architect, or perhaps you are into geometry as a discipline of logic. Here is the way to determine a square and circle of equal areas, and further, to understand the meaning of the square root of π. Use r1 to equal the side of a square and r2 to represent the radius of the corresponding circle.
Let r1^2 represent the area of the square, A(s).
Let πr2^2 = the area of the circle, A(c).
Set A(s) = A(c) via r1^2 = πr2^2.
Then r1^2 / r2^2 = π and r1 / r2 = sqrt(π).
Given either r1 or r2, we can determine the other one. That is: r1 = sqrt(π)*r2 and r2 = r1/sqrt(π). sqrt(π) = 1.77245385090552. So, given a square of side r1 = 1.77245385090552, its area = 1.77245385090552^2 = π and r2 = r1/sqrt(π) or 1.77245385090552/1.77245385090552 = 1 and the area of r2's circle = πr2^2 = π(1)^2 = π, which equals the area of the square just calculated.
And it's been learned that the square root of π means the relationship between the equal areas of a square and circle of varying "radius."
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