How to Determine a Square and Circle of Equal Area

Co-authored by Grace Imson, MA

Last Updated: February 26, 2024

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.

Part 1

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The Tutorial

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Let πr2^2 = the area of the circle, A(c).

Determine a Square and Circle of Equal Area Step 2

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."

Part 2

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Explanatory Charts, Diagrams, Photos

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Part 3

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Helpful Guidance

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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 Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category: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.
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If area of circle is equal to the area of square, then what is the relation between circumference (c) and perimeter (p) of the square?

Donagan

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The relationship is that the perimeter of the square is equal to the circumference of the circle multiplied by 1.13. Thus, p = 1.13 c. Here's how that's derived: the circle's area (πr²) is defined as being equal to the square's area (4s), where r is the circle's radius, and s is the square's side. So πr² = s², making s equal to r√π. The square's perimeter is 4s, or 4r√π. You would have to multiply the circumference (2πr) by (2/√π), or 1.13, to get the square's perimeter (4r√π). So the relationship of circumference to perimeter is p = 1.13 c. This is true only when the two areas are equal.

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What is the area of circle whose radius is the diagonal of a square whose area is 4 cm square?

Donagan

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A square with an area of 4 cm² has sides of 2 cm each and a diagonal of 2√2 cm. A circle whose radius is 2√2 cm has an area of π(2√2)² = 8π = 25.13 cm².

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If area of a square is A, how do I find the area of circle using same perimeter?

Donagan

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If the square's area is A, each side is √A. So the square's perimeter is 4√A, and so is the circle's circumference. Then the diameter of the circle is its circumference divided by pi, or (4√A) / π, meaning its radius is (2√A) / π. Then the circle's area is πr², or π(2√A / π)² = 4A / π² = (0.405)(A), or (.4A).

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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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