A centuries-old mathematical puzzle, Quadrature of the Circle (also known as “Squaring the Circle”), is thought to indicate the origin of the British system of measurements.

Squaring the circle is “the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.” (“Squaring the circle”, Wikipedia) But that is only the classical problem’s imposed limitation. Think of a plastic hula hoop (a circle) which you somehow twist and bend into a square: the area of that square would be the same as the previous hula-hoop circle.

It is generally conceded, as of 1882, that the puzzle is impossible to solve because pi (3.141592654…) is “transcendental”, i.e., not the root of any polynomial with rational coefficients. (“Squaring the circle”, Wikipedia)

However, for the purposes of his argument related to the problem of Quadrature of the Circle, J. Ralston Skinner, in 1894, begins with the caveat: the question of value of the quadrature, as to whether it is the expression of exactitude of relation, does not arise. This “work thus is relieved of any necessity of examination into the question of the possibility of what is called ‘the quadrature,’ or ‘the squaring of the circle’…” (*Key To The Hebrew-Egyptian Mystery in the Source of Measures*, by J. Ralston Skinner. 1894. Republished by www.kessinger.net)

In other words, for Mr. Skinner’s purposes, he need only *approach* an exact value. And Skinner comes quite close to this.

One Peter Metius, circa 1585 A.D., reportedly arrived at a solution giving the quadrature value of the circle. Later, in the 19th century, a John A. Parker of New York City is said to have found his own quadrature value.

The value found by Metius was 113 for diameter and 355 for circumference. Parker’s value was 6561 for diameter and 20612 for circumference. Notice how both the Metius and Parker values yield an approximation of Pi when their respective circumferences are divided by their respective diameters.

Parker’s process is a little difficult to grasp. The equilateral triangle and the circle are seen to be “opposite to one another in all the elements of their construction.” “The equilateral triangle is the primary of all shapes in nature formed of straight lines, and of equal sides and angles, and it has the least radius, the least area, and the greatest circumference of any possible shape of equal sides and angles.” (Skinner, op. cit.)

The triangle “has the least number of sides of any possible shape in nature formed in straight lines; and the circle is the ultimatum of nature in extension of the number of sides.” (ibid.)

A triangle has sides equal to 1. Totaling the three sides gives circumference 3.

3 squared multiplied by 3 squared yields 81. Two triangles joined together make a square. The perimeter of this square is 81 multiplied by 4 = 324. The area for this square would be 81 times 81 equals 6561.

“Mr. Parker makes use of an element of measure of the equilateral triangle, by which, as a least unit of measure, to express the measure of the elements of a circle in terms of the numerical value of a square: so that, as a conclusion, a square of 81 to the side, or 6561 in area, shall contain a circle whose area equals 5153; or, rectifying the circumference, a diameter of 6561 shall have a circumference of 5153 x 4 = 20612” (Skinner, op. cit.)

Suppose a cube whose total edges add up to 20612. There being 12 edges, 20612 divided by 12 yields 1717.666667. The one-thousandth part of this equals 1.71766… This number, 1.71766…, as a proportion of the British foot measurement is the ancient, Biblical cubit value.

The 20612 value also corresponds to the British inch measurement: There being 12 inches to the foot, 20612 divided by 12 yields 1717.666667.

A measurement system is discerned to be hidden in the Old Testament. In Hebrew, Jared is construed to be “the mount of descent.” Jared, in Hebrew, consists reportedly of the letters Yodh-Resh-Da’leth, which “is literally, in British, Y-R-D; hence, in Jared, is to be found, literally, our English word yard.” (Skinner, op. cit.)

Similarly, with Enoch, the son of Jared, Skinner (op. cit.) finds it noteworthy that Enoch lived 365 years. 365 is, of course, a measurement of days in the year. Skinner somehow intuits an ancient understanding of the relation between space and time.

(A version of the above first appeared at my *Melchizedek Communique* web site on December 6, 2009)

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Re: http://www.aitnaru.org/images/Pi_Corral.pdf

Having discovered with CAD software that the rPi radius must be drawn at 62.4028873643093955482677952.. degrees (and many more decimal digits), I finally learned that squaring a circle according to the Greek rules would be quite difficult.

Impossible? Advanced math, focused on the Pi ratio, proves “Yes, impossible”.

However, the complementary and precise ratio (nicknamed rPi) opens another research door and hints that all possible geometric lines, angles, and objects must be analyzed before we can truly determine that squaring the circle is impossible.

With this new perspective, we are inspired to ignore the popular transcendental Pi and search anew for geometric complements that balance the squared circle equation.

Consider this unique scalene triangle (instead of equilateral) and possibilities for squaring the circle seem intuitive, almost jumping “out of the box”. Or focus on the “Transcendence” design that suggests geometry required to prove the square.

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Re: http://www.aitnaru.org/images/Pi_Corral.pdf (see “This Way Up” design)

A non-mathematician’s review of the esoteric “This Way Up” palette:

Consider the longest lines (hypotenuse) of the two green right triangles in the squared golden circle (D = 2,000,000 units). Each triangle is a component of a larger isosceles right triangle having an hypotenuse equal to the square root of Pi; both isosceles triangles are geometrically similar.

As the hypotenuse line of the isosceles triangle (on the right of the design) rotates 45 degrees counterclockwise toward the center of the design, becoming the side of the isosceles triangle, the side of the other isosceles triangle (at the bottom of the design) rotates 45 degrees counterclockwise toward the center, becoming the hypotenuse of that isosceles triangle.

The two rotating lines have equal length (hypotenuse of green triangles) when geometric equilibrium is reached. Then, the opposite points of these lines are equidistant (red lines, equal to the square root of Pi). And the two rotated lines (hypotenuse of green triangles) have length equal to the side of a square inscribed in the golden circle.

Wow! Three years later and I’m still runnin’ in squared circles!

Here’s some of the the evidence: http://aitnaru.org/images/gEotP.pdf

Note that the “Cyan Seven Seven 16:1” geometry (near bottom of PDF) clearly invokes the circle-squaring spirit of Pythagoras. Regarding this unique circle-squaring right triangle:

Circle’s diameter is the hypotenuse of the right triangle.

2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

Diameter / 2(sqrt(1/Pi)) = long side of right triangle

= side of circle’s square. 😉

Thus, 2(sqrt(1/Pi)) is a more robust “Pi” constant because it highlights the Pythagorean geometry inherent in squared circles. Quoth the ravin’, “Furthermore …” (or at least once in a blue moon):

The long side (one side of the circle’s square) forms a scalene triangle having a 45 degree angle and a side having length equal to a side of a square inscribed in the circle.

What’s a good name for this “robust” Pi constant? Try “rPi” (for “radial Pi”) … since this constant was first explored using a circle’s radius (not its diameter). “Furthermore”, it sounds like “our Pi”, a growing perspective of squared circle aficionados.

Creating a circle and a square with equal areas involving 100% accuracy solved.

From: Liddz.

It is almost impossible to create a circle and a square with equal areas of measure with 100% accuracy due to the transcendental nature of the modern traditional version of the ratio Pi 3.141592653589793. Remember that there are also 2 other modern values of Pi in addition to the traditional most common value of Pi that is 3.141592653589793. The 2 other values of Pi are: 3.146446609406726 and also 3.14460551102969. 3.146446609406726 is the result of 14 subtract square root of 2 = 1.414213562373095 = 12.585786437626905 and then 12.585786437626905 is divided by 4. 3.14460551102969 is the result of 4 divided by the square root of the Golden ratio = 1.272019649514069. 1.272019649514069 squared is the Golden ratio of 1.618033988749895. Please remember that the Golden ratio can be obtained in Trigonometry through the formula Cosine (36) multiplied by 2. All modern Pi values must be approximated meaning the results that are given to us by the 3 modern Pi values must be reduced to 4 or 5 decimal places resulting in multiple approximations for the 3 main Pi values. There is a debate among scholars regarding which version of the 3 values of Pi are better to use when dealing with circles and geometrical figures. The view of this author is that all 3 of the values of Pi are important and all the 3 values of Pi must be studied as much as possible to achieve accuracy. The 3 values of Pi 3.141592653589793 and 3.146446609406726 and 3.144605511029693 must be compared to each other so the geometrician can determine which value is best suited for the desired task.

I think that I have solved the problem of creating a circle and a square with equal areas involving 100% accuracy:

Requirements for creating a circle and a square with equal areas involving 100% include:

1. The surface area of the circle that is to be equal to the surface area of a square can be derived from the surface area of 2 squares with rational widths of measure but does not always have to be.

2. The square that has a surface area equal to the surface area of the circle can have a width of an irrational measure or a width of rational measure.

3. The Pythagorean theorem is required to construct the square that has an irrational width of measure and a surface area equal to the surface area of the circle. The Pythagorean theorem is also required to test the accuracy of the construction of a circle plus a square that has an irrational width of measure with equal areas involving 100% accuracy. The Pythagorean theorem is based upon a scalene triangle and says that the amount of squares on the hypotenuse is equal to the sum of all the squares that are located on both the adjacent edge of the scalene triangle and also the opposite edge of the scalene triangle. The width of the square with a surface area that is equal in measure to the surface area of the circle must be equal in measure to length of the scalene triangle’s hypotenuse.

4. If the surface area of the circle is already known and the measure for the radius of the circle is not yet known a solution is to divide the surface area of the circle by Pi and then apply the result to square root. After the surface area of the circle has been divided by Pi and the result applied to square root an approximation of the circle’s radius can be given. The approximation of the circle’s radius must be reduced to 4 decimal points and 1 added to the last 4 decimal places for the approximation of the circle’s radius that has been reduced to 4 decimal places. After the approximation of the circle’s radius has been reduced to 4 decimal places 0.01 can be added to the whole value instead of adding 1 to the last of the 4 decimal digits to determine the correct measure for the circle’s radius. Other values of Pi must also be used to confirm the accuracy of the measure for the circle’s diameter. : https://en.wikipedia.org/wiki/Squaring_the_circle: http://rsjreddy.webnode.com//

http://www.jainmathemagics.com/

http://www.iosrjournals.org/iosr-jm/papers/Vol10-issue1/Version-4/C010141415.pdf

My question is it possible to create a circle with a surface area of 106 equal units because if we can create a circle with a surface area of 106 equal units of measure then we can also create a square with a surface area of 106 equal units of measure by creating a scalene triangle with the longest edge length as 9 equal units of measure taken from the diameter of the circle that has a surface area of 106 equal units of measure, while the shortest length of the scalene triangle has 5 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 9 equal units of measure and the shortest length of the scalene triangle as 5 equal units of measure is equal in measure to the width of a square that has a surface area of 106 equal units of measure. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 106 equal units of measure.

Area of circle = 106.

Diameter of circle = 11.62.

Circumference of circle = 36.5.

Traditional Pi approximated to: 3.141135972461274.

9 squared = 81.

5 squared = 25.

81 + 25 = 106.

Most values of Pi can confirm that if a circle has a diameter of 11.62 equal units of measure then the surface area of the circle with a diameter of 11.62 equal units of measure is 106 equal units of measure.

Also is it possible to create a circle geometrically on paper with a surface area of 117 square units because if we can then we can create a square with the same 117 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 9 units of measure from the circle’s diameter with an area of 117 units of equal measure. The shortest length of the scalene triangle must have 6 equal units of measure that are also taken from the circle’s diameter with an area of 117 equal square units. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has a surface area of 117 equal units of measure.

Area of circle = 117.

Diameter of circle = 12.22

Circumference of circle is 38.39.

Traditional Pi is approximated to: 3.141571194762684.

9 squared = 81.

6 squared = 36.

81 + 36 = 117.

Most values of Pi can confirm that if a circle has a diameter of 12.22 equal units of measure then the surface area of the circle with a diameter of 12.22 equal units of measure is 117 equal units of measure.

Also is it possible to create a circle geometrically on paper with a surface area of 153 square units because if we can then we can create a square with the same 153 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 12 units of measure from the circle’s diameter with an area of 153 units of equal measure. The shortest length of the scalene triangle must have 3 equal units of measure that are also taken from the circle’s diameter with an area of 153 equal square units. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has a surface area of 153 equal units of measure.

Area of circle = 153.

Diameter of circle = 13.96

Circumference of circle is 43.85.

Traditional Pi is approximated to: 3.141117478510029.

12 squared = 144.

3 squared = 9.

144 + 9 = 153.

Most values of Pi can confirm that if a circle has a diameter of 13.96 equal units of measure then the surface area of the circle with a diameter of 13.96 equal units of measure is 153 equal units of measure.

Also is it possible to create a circle geometrically on paper with a surface area of 365 square units because if we can then we can create a square with the same 365 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 14 units of measure from the circle’s diameter with an area of 365 units of equal measure. The shortest length of the scalene triangle must have 14 equal units of measure that are also taken from the circle’s diameter with an area of 365 equal square units. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has a surface area of 365 equal units of measure.

Area of circle = 365.

Diameter of circle = 21.56

Circumference of circle is 67.73.

Traditional Pi is approximated to: 3.141465677179963.

14 squared = 196.

13 squared = 169.

196 + 169 = 365.

Most values of Pi can confirm that if a circle has a diameter of 21.56 equal units of measure then the surface area of the circle with a diameter of 21.56 equal units of measure is 365 equal units of measure.

Also is it possible to create a circle with a surface area of 530 equal units because if we can create a circle with a surface area of 530 equal units of measure then we can also create a square with a surface area of 530 equal units of measure by creating a scalene triangle with the longest edge length as 19 equal units of measure taken from the diameter of the circle that has a surface area of 530 equal units of measure, while the shortest length of the scalene triangle has 13 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 19 equal units of measure and the shortest length of the scalene triangle as 13 equal units of measure is equal in measure to the width of a square that has a surface area of 530 equal units of measure. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 530 equal units of measure.

Area of circle = 530.

Diameter of circle = 25.98.

Circumference of circle = 81.61.

Traditional Pi approximated to: 3.141262509622787.

19 squared = 361.

13 squared = 169.

361 + 169 = 530.

Most values of Pi can confirm that if a circle has a diameter of 25.98 equal units of measure then the surface area of the circle with a diameter of 25.98 equal units of measure is 530 equal units of measure.

Also is it possible to create a circle with a surface area of 612 equal units because if we can create a circle with a surface area of 612 equal units of measure then we can also create a square with a surface area of 612 equal units of measure by creating a scalene triangle with the longest edge length as 24 equal units of measure taken from the diameter of the circle that has a surface area of 530 equal units of measure, while the shortest length of the scalene triangle has 6 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 24 equal units of measure and the shortest length of the scalene triangle as 6 equal units of measure is equal in measure to the width of a square that has a surface area of 612 equal units of measure. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 612 equal units of measure.

Area of circle = 612.

Diameter of circle = 27.92.

Circumference of circle = 87.71.

Traditional Pi approximated to: 3.14147564469914.

24 squared = 576.

6 squared = 36.

576 + 36 = 612.

Most values of Pi can confirm that if a circle has a diameter of 27.92 equal units of measure then the surface area of the circle with a diameter of 27.92 equal units of measure is 612 equal units of measure.

Also Is it possible to create a circle with a surface area of 36 equal units because if we can create a circle with a surface area of 36 equal units of measure then we can also create a square with a surface area of 36 equal units of measure by creating a square with 6 equal units of measure that are derived from the diameter of the circle with a surface area of 36 equal units of measure. The Pythagorean theorem is not required to construct a square with a surface area equal to the surface area of a circle with 36 equal units of measure. 6 squared is 36. The ratio of both the diameter of the circle divided by the width of the square and also the ratio of the radius of the circle divided by half the width of the square must be 1.13. The hypotenuse of a scalene triangle with its second longest length as 3.39 equal units of measure while the shortest length of the scalene triangle is 3 equal units of measure has a measuring angle of 48.49 degrees when the second longest length of the scalene triangle is vertical and 41.50 degrees when the second longest length of the scalene triangle is horizontal.

Area of circle = 36

Diameter of the circle is 6.78 equal units of measure.

Circumference of the circle is 21.3 equal units of measure.

Traditional Pi approximated to: 3.141592920353982.

6 squared is 36.

Most values of Pi can confirm that if a circle has a diameter of 6.78 equal units of measure then the surface area of the circle with a diameter of 6.78 equal units of measure is 36 equal units of measure.

Also Is it possible to create a circle with a surface area of 144 equal units because if we can create a circle with a surface area of 144 equal units of measure then we can also create a square with a surface area of 144 equal units of measure by creating a square with 12 equal units of measure that are derived from the diameter of the circle with a surface area of 144 equal units of measure. The Pythagorean theorem is not required to construct a square with a surface area equal to the surface area of a circle with 36 equal units of measure. 12 squared is 144. The ratio of both the diameter of the circle divided by the width of the square and also the ratio of the radius of the circle divided by half the width of the square must be 1.13. The hypotenuse of a scalene triangle with its second longest length as 6.78 equal units of measure while the shortest length of the scalene triangle is 6 equal units of measure has a measuring angle of 48.49 degrees when the second longest length of the scalene triangle is vertical and 41.50 degrees when the second longest length of the scalene triangle is horizontal.

Area of circle = 144

Diameter of the circle is 13.56 equal units of measure.

Circumference of the circle is 42.6 equal units of measure.

Traditional Pi approximated to: 3.141592920353982.

12 squared is 144.

Most values of Pi can confirm that if a circle has a diameter of 13.56 equal units of measure then the surface area of the circle with a diameter of 13.56 equal units of measure is 144 equal units of measure.

Circle and square with equal areas or approximate equal areas.

Squaring the circle by creating a circle and a square with equal areas:

Squaring the circle involves creating a circle with a circumference equal to the perimeter of a square. Also squaring the circle can involve creating a circle and a square with equal areas or approximate equal areas. Squaring the circle can also include harmonious relationships such as the part of the square that intersects the circle’s circumference can be similar to the radius of the circle or the same as the radius of the circle or equal to half of the square’s edge length. Squaring the circle with the area of the square being equal to the area of the circle usually cannot be achieved with 100% accuracy because traditional Pi 3.141592653589793 has been proven to be Transcendental in addition to being irrational. Traditional Pi 3.141592653589793 is Transcendental because Traditional Pi 3.141592653589793 does not fit any polynomial equations. Squaring the circle becomes possible and easy after traditional Pi 3.141592653589793 has been rejected and replaced with other values of Pi that are NOT transcendental. Golden Pi = 3.144605511029693 is irrational but Golden Pi is NOT transcendental because Golden Pi = 3.144605511029693 is the only value of Pi that fits the following polynomial equations: 8th degree polynomial for Golden Pi: π8 + 16π6 + 163π2 = 164.

4th dimensional equation/polynomial for Golden Pi = 3.144605511029693 (x4 + 16×2 – 256 = 0).

A polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

Both Golden Pi = 3.144605511029693 and Pi accepted as 22 divided by 7 = 3.142857142857143 can be used to create a circle and a square with equal areas of measure involving 100% accuracy.My question is it possible to create a circle with a surface area of 106 equal units because if we can create a circle with a surface area of 106 equal units of measure then we can also create a square with a surface area of 106 equal units of measure by creating a scalene triangle with the second longest edge length as 9 equal units of measure taken from the diameter of the circle that has a surface area of 106 equal units of measure, while the shortest length of the scalene triangle has 5 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 9 equal units of measure and the shortest length of the scalene triangle as 5 equal units of measure is equal in measure to the width of a square that has a surface area of 106 equal units of measure. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene right triangle also called the hypotenuse also has 106 equal units of measure.

Area of circle = 106.

Rational measure for the diameter of circle = 11.62.

Irrational measure for the diameter of the circle = 11.61180790611399 according to Golden Pi = 3.144605511029693.

Irrational measure for the diameter of the circle = 11.61180790611399 divided by the width of the square the square root of 106 = the square root of the Golden root = 1.127838485561683.

The Golden root = 1.272019649514069.The Golden root = 1.272019649514069 is the square root of Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

Irrational measure for the circumference of the circle = 36.514555134584213 according to Golden Pi = 3.144605511029693.

Square root of Golden Pi = 3.144605511029693 = 1.773303558624324

9 squared = 81.

5 squared = 25.

81 + 25 = 106.

Most values of Pi can confirm that if a circle has a rational measure for the diameter as 11.62 equal units of measure then the surface area of the circle with a rational measure for the diameter of 11.62 equal units of measure is 106 equal units of measure.

“The ancient Egyptian square root of Pi rectangle and the ancient Egyptian square root for the Golden root rectangle and also the Pythagorean theorem”: https://en.wikipedia.org/wiki/Pythagorean_theorem

Is it possible to create a circle with a surface area of 154 equal units because if we can create a circle with a surface area of 154 equal units of measure then we can also create a square with a surface area of 154 equal units of measure by creating a scalene triangle with the second longest edge length as 12 equal units of measure taken from the diameter of the circle that has a surface area of 154 equal units of measure, while the shortest length of the scalene triangle has 4 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 12 equal units of measure and the shortest length of the scalene triangle as 4 equal units of measure is equal in measure to the width of a square that has a surface area of 154 equal units of measure. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene right triangle also called the hypotenuse also has a surface area of 154 equal units of measure.

Area of circle = 154.

Diameter of circle = 14.

Circumference of circle = 44.

Ancient Egyptian Pi = 22 divided by 7 = 3.142857142857143.

12 squared = 144.

3 squared = 9.

1 squared = 1

144 + 9 + 1 = 154.

A square with a surface area of 154 equal units of measure can be created if the second longest edge length of a scalene right triangle has 12 equal units of measure while the shortest edge length of the scalene right triangle has 4 equal units of measure. According to the Pythagorean theorem if a square has a width that is equal to the hypotenuse of a scalene right triangle that has its second longest edge length as 12 equal units of measure while the shortest edge length for the scalene right triangle has 4 equal units of measure then the surface area of the square that has a width equal to the measure of the hypotenuse for the scalene right triangle that has its second longest edge length as 12 while its shortest edge length is 4 is 154 equal units of measure. If the width of the square that has a surface area of 154 equal units of measure is then accepted as the longer length of a square root of ancient Egyptian Pi = 1.772810520855837 rectangle then a circle can be created with the shorter edge length of the square root of ancient Egyptian Pi = 1.772810520855837 rectangle being equal in measure to the radius of the circle with a surface area equal to the surface area of the square that has a surface are of 154 equal units of measure. According to ancient Egyptian Pi = 3.142857142857143 if the radius of a circle has 7 equal units of measure then the surface are of the circle is 154 equal units of measure. The measuring angles for a square root of ancient Egyptian Pi = 1.772810520855837 rectangle are 60.57369496075449 degrees and 29.42630503924551 degrees. 60.57369496075449 degrees can be gained when the square root of ancient Egyptian Pi = 1.772810520855837 is applied to the inverse of Tangent in Trigonometry. 29.4263050392455 degrees can be gained when the ratio 0.564076074817766 is applied to the inverse of Tangent in Trigonometry. If a circle with a diameter of 14 equal units of measure has already been created so that the surface area of the circle can have 154 equal units of measure according to ancient Egyptian Pi = 3.142857142857143 and the desire is to have a square that also has a surface area equal to the circle’s surface area of 154 equal units of measure then a solution is to add 1 quarter of the circle’s circumference hat is 11 to the diameter of the circle with 14 equal units of measure and at the division point where 14 is subtracted from the diameter line of 25 equal units of measure draw right angles that can touch the circumference of a circle or a semi-circle if the diameter of 25 equal units of measure is divided into 2 halves. A rectangle with its longest length as 14 while its second longest length is the square root of 154 has the ratio for the square root of the Golden root as 1.128152149635533. 1.128152149635533 is the square root of 1.272727272727273. 4 divided by 1.272727272727273 is ancient Egyptian Pi = 3.142857142857143. So the longer length of the ancient Egyptian square root for the Golden root = 1.128152149635533 rectangle is 14, the diameter of the circle with a surface area of 154, while the shorter length of the ancient Egyptian square root for the Golden root = 1.128152149635533 rectangle is the square root of 154 = 12.40967364599086, the width of the square.

1.128152149635532 squared is 1.272727272727272 and 1.272727272727272 squared is the Golden ratio of 1.619834710743799. The ancient Egyptian square root for the Golden root = 1.128152149635532 is important.

Rules for squaring the circle with equal areas, the creation of a circle and square with equal areas involving 100% accuracy:

1. The surface area of the circle that is to be equal to the surface area of a square can be derived from the surface area of 2 squares with rational widths of measure.

2. The square that has a surface area equal to the surface area of the circle can have a width of an irrational measure.

3. The Pythagorean theorem can be used to construct the square that has an irrational width of measure and a surface area equal to the surface area of the circle. The Pythagorean theorem can also be used to test the accuracy of the construction of a circle plus a square that has an irrational width of measure with equal areas involving 100% accuracy. The Pythagorean theorem is based upon a right triangle and says that the amount of squares on the hypotenuse is equal to the sum of all the squares that are located on both the second longest edge of the scalene right triangle and also the shortest edge of the scalene triangle.

If the measurements for the second longest and shortest edge lengths of a scalene right triangle are already known and the measuring angle of the hypotenuse is already known and the desire is to know the measure for the hypotenuse of the scalene right triangle an alternative to the Pythagorean theorem is to divide the measure of the second longest edge length or the measure of the shortest edge length by the ratio that is gained from applying the measuring angle of the hypotenuse of the scalene triangle to the Sine function in trigonometry. Applying the measuring angle of the hypotenuse of a scalene triangle to the Sine function in trigonometry and then dividing the second longest edge length or the shortest edge length by the ratio that is gained from applying the measuring angle of the hypotenuse of a scalene triangle to the Sine function in trigonometry so that the length of the hypotenuse of the scalene triangle can be known is further confirmation regarding the proof of the theorem of Pythagoras. If the measurements for the second longest and shortest edge lengths of a scalene right triangle are already known and the measuring angle of the hypotenuse of the scalene triangle is already known and the desire is to know the measure for the hypotenuse of the scalene right triangle another alternative to the Pythagorean theorem is to divide the measure of the second longest edge length or the measure of the shortest edge length of the scalene right triangle by the measuring angle of the hypotenuse through the Cosine function in Trigonometry. Dividing the second longest edge length or the shortest edge length by the measuring angle of the hypotenuse of a scalene triangle to the Cosine function in trigonometry so that the length of the hypotenuse can be known is further confirmation regarding the proof of the theorem of Pythagoras. Remember that a scalene right triangle can also be half of a rectangle.

4. If the edge of the square is the longer length of a square root of Pi rectangle while the radius of the circle is the shorter length of the square root of Pi rectangle then irrational measure for the radius of the circle can be gained and does not have to be approximated. If the diameter of the circle is the longer length of a 1.127838485561682 ratio rectangle while the width of the square is the shorter length of the 1.127838485561682 ratio rectangle then the irrational measure for the radius of the circle can be gained and does not have to be approximated. Remember that the radius of a circle may have 2 measures: 1. The irrational measure that was gained by the chosen value of Pi. 2: The rational measure that can be confirmed by most values of Pi.

Alternatively if the surface area of the circle is already known and the measure for the radius of the circle is not yet known a solution is to divide the surface area of the circle by Pi and then apply the result to square root. After the surface area of the circle has been divided by Pi and the result applied to square root an approximation of the circle’s radius can be given. The approximation of the circle’s radius must be reduced to 4 decimal points and 1 added to the last 4 decimal places for the approximation of the circle’s radius that has been reduced to 4 decimal places. After the approximation of the circle’s radius has been reduced to 4 decimal places 0.01 can be added to the whole value instead of adding 1 to the last of the 4 decimal digits to determine the correct measure for the circle’s radius. Other values of Pi must also be used to confirm the accuracy of the rational measure for the circle’s diameter.

The radius of the circle multiplied by the square root of Pi results in the width of the square that has a surface area equal to the surface area of the circle.

5. A square root of Pi rectangle can help to create a circle and a square with equal areas of measure involving 100% accuracy if the longer edge of the square root of Pi rectangle is interpreted to be the width of the square while the shorter edge of the square root of Pi rectangle is interpreted to the measure for the radius of the circle.

6. A rectangle based upon the square root for the square root of the Golden ratio = 1.127838485561682 can help to create a circle and a square with equal areas of measure if the longer edge of the square root for the square root of the Golden ratio = 1.127838485561682 rectangle is interpreted as the diameter of the circle. The shorter edge of the square root for the square root of the Golden ratio = 1.127838485561682 rectangle must be interpreted as the width of the square in relation to the longer edge of the square root for the square root of the Golden ratio = 1.127838485561682 rectangle being interpreted as the diameter of the circle with a surface area equal to the square that has a width equal in measure to the shorter edge length of the square root for the square root of the Golden ratio = 1.127838485561682 rectangle. The diameter of a semi-circle or a circle can be divided into the square root for the square root of the Golden ratio = 1.127838485561682 if the length of the semi-circle’s diameter or the circle’s diameter was created from the shortest edge length of a Kepler right scalene triangle being added to the measure of the second longest edge length of the Kepler right scalene triangle. The square root for the square root of the Golden ratio = 1.127838485561682 is also known as the square root of the Golden root.

7. If a circle and a square are created with equal areas of measure and the perimeter of the square is divided by the circumference of the circle the resulting ratio will be 1.127838485561682 if Golden Pi = 3.144605511029693 is used to calculate the surface area of the circle or 1.128152149635533 if 22 divided by 7 = 3.142857142857143 is used as Pi to calculate the surface area of the circle. Remember that 1.127838485561682 is the square root of the Golden root while 1.128152149635533 is an approximation for the square root of the Golden root. The Golden root is 1.27201964951406 and 1.27201964951406 is the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895. If a circle and a square are created with equal areas of measure and the perimeter of the square is divided by the diameter of the circle the resulting ratio will be 3.546607117248614. The square root of Golden Pi = 1.773303558624307 is half of the ratio 3.546607117248614.

Rules for squaring the circle with equal areas, the creation of a circle and square with equal areas involving 100% accuracy:8. If a circle and a square are created with the same surface area and then the surface area of the circle or the square is divided by the surface area of a smaller circle that has a radius equal in measure to the half the width of the square that has a surface area equal in measure to the larger circle the resulting ratio is the square root of the Golden ratio also called the Golden root = 1.27201964951406. So if the surface area of the circle is known but the width of the square with the same surface area is not yet known another solution is to divide the surface are of the circle by the square root of the Golden ratio also called the Golden root = 1.27201964951406. The result is the surface area of a circle with a radius equal in measure to half the width of the square that has a surface area equal to the surface area of the original circle. To know the measure for the width of the square that has a surface area equal to the surface area of the original larger circle divide the surface area of the smaller circle that was determined by the Golden root = 1.27201964951406 by Golden Pi = 3.144605511029693 and the result of dividing the surface area of the smaller circle by Golden Pi = 3.144605511029693 must then be applied to square root revealing half the width of the square that has a surface area equal to the surface area of the larger original circle.

9. If the diagonal of a square root of the Golden root 1.127838485561682 rectangle is divided by the shorter edge of the square root of the Golden root 1.127838485561682 rectangle then the result is the ratio 1.507322012548768. 1.507322012548768 can be reduced to 1.5. The ratio 1.507322012548768 is approximately 1.5, one and a half of one.

The ratio 1.507322012548768 can be gained if 1 is added to the square root of the Golden ratio also called the Golden root = 1.27201964951406. 1.27201964951406 plus 1 = 2.27201964951406. The square root of 2.27201964951406 = 1.507322012548768. Remember that the Golden ratio can be gained through Cosine (36) multiplied by 2 = 1.618033988749895 in Trigonometry.

10. The surface area of an isosceles triangle that is made from 2 Kepler right triangles is the same measure as the radius squared for a circle with a surface area equal to the surface area of a square with a width that is equal to the base length of the isosceles triangle that is made from 2 Kepler right triangles. The surface area of an isosceles triangle that is made from 2 Kepler right scalene triangles is the same as the surface area of a circle with a diameter equal to the height of the isosceles triangle that is made from 2 Kepler right scalene triangles. If a circle is created with the same surface area as a square that has a width equal to the base width of a isosceles triangle that is made from 2 Kepler right triangles then the radius of the circle is also equal to the width of a square that has a surface area equal to the surface area of a circle with a diameter equal in measure to the height of the isosceles triangle that is made from 2 Kepler right triangles. If an isosceles triangle is created from 2 Kepler right scalene triangles and the base width of the isosceles triangle that is made from 2 Kepler right triangles is divided by the height of the isosceles triangle that is made from 2 Kepler right triangles then the result is half of the irrational ratio Pi. If a isosceles triangle that is made from 2 Kepler right triangles is inscribed in a circle and the surface area of the circle is divided by the surface area of the isosceles triangle that is made from Kepler right triangles that has been inscribed in the circle the result is the Golden ratio squared = 2.618033988749895.

11. The surface area of a squared divided by the surface area of circle with a diameter equal to the width of the square is 1.27201964951406. 1.27201964951406 is the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

12. The ratio for the surface area of a circle divided by the surface area of square that has its diagonal equal in measure as the diameter of the circle is half of Golden Pi = 1.572302755514847. 2 divided by 1.27201964951406 the square root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895 is half of Golden Pi = 1.572302755514847.