Determine a Square and Circle of Equal Area

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.

Steps

The Tutorial

1. Let r1^2 represent the area of the square, A(s).
2. Let πr2^2 = the area of the circle, A(c).
3. Set A(s) = A(c) via r1^2 = πr2^2.
4. Then r1^2 / r2^2 = π and r1 / r2 = sqrt(π).
5. 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.
6. 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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Helpful Guidance

1. Make use of helper articles when proceeding through this tutorial:
   • See the article How to Determine a Square and Circle of Equal Perimeter for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
   • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Tips

• This also may mean that the "entire variable radius" of the square is equal to the standard radius of the equal circle, during a 360 turn, but I have yet to prove that. I think it may be unequal if the first derivatives are unequal.

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Sources and Citations

• The source workbook for this article is "AREAS equal.xlsx".

See also

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