**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.