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Circle and square with equal areas or approximate equal areas.

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.

[en.wikipedia.org]:

[rsjreddy.webnode.com]

[www.jainmathemagics.com]

[www.iosrjournals.org]

It is also usually impossible to create a circle and a square with equal areas involving 100% accuracy because if the area of the circle is rational then the width of the square will be irrational and the opposite is also true meaning that if the width of the square is rational then the area of the circle will be irrational. There are numerous examples of creating a circle and a square with equal areas involving 100% accuracy but only 11 examples for the creation of a circle and a square with equal areas involving 100% accuracy is included here.

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.

“The first example of creating a circle and square with equal areas of measure involving 100% accuracy”:

Is it possible to create a circle geometrically on paper with a surface area of 45 square units because if we can then we can create a square with the same 45 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 6 units of measure from the circle’s diameter with an area of 45 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 surface area of 45 equal square units. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 45 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 triangle also called the hypotenuse also has a surface area of 45 equal units of measure.

Area of circle = 45.

Diameter of circle = 7.58.

Circumference of circle is 23.81.

Traditional Pi is approximated to: 3.141160949868074.

6 squared = 36.

3 squared = 9.

36 + 9 = 45.

Most values of Pi will confirm that if a circle has a diameter of 7.58 equal units of measure then the surface area is 45 equal units of measure.

“The second example of creating a circle and 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 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 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.

“The third example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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.

“The fourth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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 surface area of 153 equal square units. We can use the theorem of Pythagoras: 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 will confirm that if a circle has a diameter of 13.96 equal units of measure then the surface area is 153 equal units of measure.

“The fifth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

Also is it possible to create a circle geometrically on paper with a surface area of 313 square units because if we can then we can create a square with the same 313 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 13 units of measure from the circle’s diameter with an area of 313 units of equal measure. The shortest length of the scalene triangle must have 12 equal units of measure that are also taken from the circle’s diameter with an area of 313 equal square units. We can use the theorem of Pythagoras: 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 313 equal units of measure.

Area of circle = 313.

Diameter of circle = 19.98

Circumference of circle is 62.76.

Traditional Pi is approximated to: 3.141141141141141.

13 squared = 169.

12 squared = 144.

169 + 144 = 313.

Most values of Pi can confirm that if a circle has a diameter of 19.98 equal units of measure then the surface area of the circle with a diameter of 19.98 equal units of measure is 313 equal units of measure.

“The sixth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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.

“The seventh example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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 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.

“The eighth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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.

“The ninth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

Is it possible to create a circle with a surface area of 720 equal units because if we can create a circle with a surface area of 720 equal units of measure then we can also create a square with a surface area of 720 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 720 equal units of measure, while the shortest length of the scalene triangle has 12 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 25 equal units of measure and the shortest length of the scalene triangle as 12 equal units of measure is equal in measure to the width of a square that has a surface area of 720 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 triangle also called the hypotenuse also has a surface area of 720 equal units of measure.

Area of circle = 720.

Diameter of circle = 30.28.

Circumference of circle = 95.12.

Traditional Pi approximated to: 3.141347424042272.

24 squared = 576.

12 squared = 144.

576 + 144 = 720.

Most values of Pi can confirm that if a circle has a diameter of 30.28 equal units of measure then the surface area of the circle with a diameter of 30.28 equal units of measure is 720 equal units of measure.

“The tenth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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.

“The tenth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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.

1 of the most accurate methods for creating a circle and a square with approximate equal square areas of measure is to create a circle with a diameter of 13 equal units of measure, while the width of the square has 11.52 equal units of measure.

Modern Pi = 3.141592653589793. Modern pi approximated to 40.84 divided by 13 = 3.141538461538462 is the correct approximation of modern Pi to use for this equation. Modern Pi approximated to 3.141538461538462 multiplied by 6.5 equal units of measure the radius of the circle = 20.42. 20.42 multiplied by the radius of the circle 6.5 equal units of measure = 132.73.

So the area of the circle is 132.73.

The width of the square is 11.52 equal units of measure.

11.52 multiplied by 11.52 or 11.52 squared = 132.7104.

So if the results are reduced to 4 decimal places then the area of the circle and the area of the square can be the same.

The width of the square is 11.52 equal units of measure and 11.52 multiplied by 11.52 = 132.7104. 11.52 squared = 132.7104. So the area of the circle is close to the width of the square according to the modern version of Pi approximated to 3.141538461538462. The diameter of the circle with 13 equal units of measure divided by the width of the square with 11.52 equal units of measure produces the irrational ratio of 1.128472222222222. The radius of the circle with area of 132.7104 square units of measure again is 6.5 equal units of measure so if a rectangle is created with the longest length of the rectangle being 6.5 equal units of measure while the shortest length of the rectangle is 5.76 equal units of measure the diagonal of the rectangle will measure 48.45 degrees when the longest length of the rectangle is vertical. Regarding the creation of a rectangle with the longest length measuring 6.5 equal units of measure while the shortest length measures 5.76 units of measure the diagonal of the rectangle will measure 41.55 degrees when the longest length of the rectangle is horizontal and the shortest length of the rectangle is vertical. Knowing the angle that is formed between the 2 poles of the circle’s diameter and the centre edge of the square that is parallel to the diameter of the circle can make the creation of a circle and square with approximate equal areas quicker and easier.

If a scalene triangle is created with the second longest length being 6.5 equal units of measure while the shortest length of the scalene triangle is 5.76 equal units of measure then the longest length of the scalene triangle also called the hypotenuse is 8.68 equal units of measure. The ratio of a scalene triangle’s hypotenuse that measures 8.68 equal units of measure divided by the shortest length of the scalene triangle 5.76 equal units of measure is 1.506944444444444

So if a square and a circle have approximate equal areas then the measure for the diameter divided by the width of the square can produce the ratios 1.128472222222222 or 1.129 plus an infinite amount of decimal places or 1.125 if the diameter of the circle is 9 and the width of the square is 8. 1.130434782608696 is the irrational ratio that can be obtained when the diameter of a circle has 26 equal units of measure and is divided by the width of a square with 23 equal units of measure. 26 divided by 23 = 1.130434782608696.

To make that above statement again clearer if a circle and a square are created with approximate equal areas of measure then when the measure for the diameter of the circle is divided by the measure for the width of the square then the result will be any of the following ratios depending on the amount of accuracy that is used:

1.13.

1.128 plus an infinite amount of decimal places.

1.129 plus an infinite amount of decimal places.

1.130 plus an infinite amount of decimal places.

1.125 when the diameter of the circle is 9 or a multiple of 9 and the width of the square is 8 or a multiple of 8.

“Creating a circle and a square with approximate equal areas from whole numbers”:

To obtain the area of a circle any version of Pi must be multiplied by the radius of the circle and the result of Pi being multiplied by the radius of the circle must then be multiplied by the radius of the circle for the measure of the circle’s area to be known. So in this case that is 22 divided by 7 this approximation version of Pi = 3.142857142857143 multiplied by 4.5 = 14.142857142857143 multiplied by 4.5. = 63.642857142857143. So the area of a circle with a diameter of 9 equal units of measure according to the ancient approximation version of Pi 22 divided by 7 = 3.142857142857143 is 63.642857142857143.

The area of a square can be known by multiplying the width of the square by the width of the square. So in this case the width of the square has 8 equal units of measure and 8 multiplied by 8 = 64. 8 squared = 64. So the area of a square with a width of 8 equal units of measure is 64 equal units of measure. So the area of the circle is 63.642857142857143 equal units of measure, while the area f the square is 64 equal units of measure, when a square and a circle is created to have equal areas and the diameter of the circle has 9 equal units of measure while the width of the square has 8 equal units of measure. Both the circle and the square must share the same centre. So the area of the circle is 0.36 % away from the measure for the area of the square according to the approximation version of the modern version of Pi that = 3.142857142857143. So when the values of the areas of both the circle and square are read without an infinite amount of decimal places for the area of the circle the area of the circle read 63.64 equal units of measure while the area of the square reads 64 equal units of measure.

Again the width of the square has 8 equal units of measure and 8 multiplied by 8 = 64. 8 squared = 64. So the area of a square with a width of 8 equal units of measure is 64 equal units of measure and if the radius of the circle has 4.5 equal units of measure the diameter of the circle has 9 equal units of measure. The circumference of a circle with a diameter of 9 equal units of measure is 28.27 units of measure. So Pi can be approximated to 3.141111111111111. If Pi approximated to 3.141111111111111 is multiplied by the radius of the circle the result is 14.135 and 14.135 multiplied by 4.5 the radius of circle says that the area of a circle with a diameter of 9 equal units of measure is 63.6075.

If a scalene triangle is created with the second shortest length being equal to 13 equal units of measure while the shortest length is equal to 11.52 equal units of measure then the longest length is the hypotenuse and that measures 17.36 equal units of measure. 17.36 divided by 11.52 also equals 1.506944444444444.

“Another version for creating a circle and a square with approximate equal areas”.

The following version for creating a circle and a square with approximate equal areas involves the diameter of the circle having 7 equal units of measure while the width of the square has 6.2 equal units of measure plus the 22 divided by 7 approximation of modern Pi. 22 dived by 7 = 3.142857142857143.

So in this case 22 divided by 7 = 3.142857142857143 multiplied by 3.5 the radius of the circle = 11 multiplied by 3.5 the radius of the circle = 38.5.

The width of the square is 6.2 equal units of measure. 6.2 squared = 38.44 equal units of measure. Also 6.2 multiplied by 6.2 = 38.44 equal units of measure. So again the width of the square is 6.2 equal units of measure and a square with a width of 6.2 equal units of measure has an area of 38.44 units of measure. The radius of the circle is 3.5 equal units of measure and a circle with a diameter of 7 equal units of measure has a radius of 3.5 equal units of measure and again a circle with a radius of 3.5 equal units of measure has an area of 38.5 equal units of measure according to the approximation value of modern Pi that is 22 divided by 7 = 3.142857142857143. Regarding the third version for the creation of a circle and a square with equal areas the area of the circle is 0.06 larger than the area of the square and that means that the measure for both the area of the circle and the square are very close.

“Creating a Circle and square with equal areas from existing measurements of a circle and a square with equal perimeters”:

If a circle with a circumference equal to the perimeter of a square has already been created and the new desire is to create a square with an area approximately equal to the area of the existing circle that has a circumference equal to the perimeter of another square then a solution is to obtain the ratio of the diameter of the circle divided by the edge length of the square with a perimeter equal to the circumference of the circle with the appropriate diameter and then apply the result of the circle’s diameter being divided by the edge of the square with a perimeter equal to the circumference of the circle to square root and the result is the ratio of the diameter of the existing circle divided by the edge length of a square with an area approximately equal to the area of the existing circle.

An Example follows:

1. Measurements of a circle and square with equal perimeters derived from Pi approximated to 3.141361256544503: Diameter of circle is 11.46 equal units of measure and edge length of square is 9 equal units of measure. Perimeter of square is 36 and circumference of circle is 36.

The ratio of the diameter of the circle divided by the edge length of the square is 1.273333333333333 and can also be obtained if 4 is divided by Pi approximated to 3.141361256544503 = 1.273333333333333.

2. If the ratio 1.273333333333333 is applied to square root meaning the square root of 1.273333333333333 = 1.128420725320717.

3. The ratio 1.128420725320717 can tell us the measure for the correct edge length of the square with an area approximately equal to the area of a circle with a diameter of 11.46 equal units of measure. A square with an edge length of 10.155786527886455 equal units of measure is approximately equal to the area of a circle with a diameter that has 11.46 equal units of measure.

4. 10.155786527886455 multiplied by 10.155786527886455 = 103.14. 10.155786527886455 squared = 103.14. Pi approximated to 3.141361256544503 multiplied y 5.73 equal units of the measure being the radius of the circle = 18 and 18 multiplied by 5.73 = 103.14. 103.14 is the area of a circle with a diameter of 11.46 equal units of measure according to Pi approximated to 3.141361256544503. Remember that the measure for the edge of the square must be reduced to 4 decimal places so that means that it is usually impossible for the area of the square and the area of the circle to have exactly the same measure of square units. The edge length of the square reduced to 4 decimal places is 10.15 and 10.15 squared is 103.225. So both the area of the square and the circle are quite close but not exactly the same in measure.

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.

[en.wikipedia.org]:

[rsjreddy.webnode.com]

[www.jainmathemagics.com]

[www.iosrjournals.org]

It is also usually impossible to create a circle and a square with equal areas involving 100% accuracy because if the area of the circle is rational then the width of the square will be irrational and the opposite is also true meaning that if the width of the square is rational then the area of the circle will be irrational. There are numerous examples of creating a circle and a square with equal areas involving 100% accuracy but only 11 examples for the creation of a circle and a square with equal areas involving 100% accuracy is included here.

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.

“The first example of creating a circle and square with equal areas of measure involving 100% accuracy”:

Is it possible to create a circle geometrically on paper with a surface area of 45 square units because if we can then we can create a square with the same 45 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 6 units of measure from the circle’s diameter with an area of 45 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 surface area of 45 equal square units. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 45 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 triangle also called the hypotenuse also has a surface area of 45 equal units of measure.

Area of circle = 45.

Diameter of circle = 7.58.

Circumference of circle is 23.81.

Traditional Pi is approximated to: 3.141160949868074.

6 squared = 36.

3 squared = 9.

36 + 9 = 45.

Most values of Pi will confirm that if a circle has a diameter of 7.58 equal units of measure then the surface area is 45 equal units of measure.

“The second example of creating a circle and 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 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 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.

“The third example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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.

“The fourth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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 surface area of 153 equal square units. We can use the theorem of Pythagoras: 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 will confirm that if a circle has a diameter of 13.96 equal units of measure then the surface area is 153 equal units of measure.

“The fifth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

Also is it possible to create a circle geometrically on paper with a surface area of 313 square units because if we can then we can create a square with the same 313 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 13 units of measure from the circle’s diameter with an area of 313 units of equal measure. The shortest length of the scalene triangle must have 12 equal units of measure that are also taken from the circle’s diameter with an area of 313 equal square units. We can use the theorem of Pythagoras: 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 313 equal units of measure.

Area of circle = 313.

Diameter of circle = 19.98

Circumference of circle is 62.76.

Traditional Pi is approximated to: 3.141141141141141.

13 squared = 169.

12 squared = 144.

169 + 144 = 313.

Most values of Pi can confirm that if a circle has a diameter of 19.98 equal units of measure then the surface area of the circle with a diameter of 19.98 equal units of measure is 313 equal units of measure.

“The sixth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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.

“The seventh example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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 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.

“The eighth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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: 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.

“The ninth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

Is it possible to create a circle with a surface area of 720 equal units because if we can create a circle with a surface area of 720 equal units of measure then we can also create a square with a surface area of 720 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 720 equal units of measure, while the shortest length of the scalene triangle has 12 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 25 equal units of measure and the shortest length of the scalene triangle as 12 equal units of measure is equal in measure to the width of a square that has a surface area of 720 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 triangle also called the hypotenuse also has a surface area of 720 equal units of measure.

Area of circle = 720.

Diameter of circle = 30.28.

Circumference of circle = 95.12.

Traditional Pi approximated to: 3.141347424042272.

24 squared = 576.

12 squared = 144.

576 + 144 = 720.

Most values of Pi can confirm that if a circle has a diameter of 30.28 equal units of measure then the surface area of the circle with a diameter of 30.28 equal units of measure is 720 equal units of measure.

“The tenth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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.

“The tenth example of creating a circle and square with equal areas of measure involving 100% accuracy”:

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.

1 of the most accurate methods for creating a circle and a square with approximate equal square areas of measure is to create a circle with a diameter of 13 equal units of measure, while the width of the square has 11.52 equal units of measure.

Modern Pi = 3.141592653589793. Modern pi approximated to 40.84 divided by 13 = 3.141538461538462 is the correct approximation of modern Pi to use for this equation. Modern Pi approximated to 3.141538461538462 multiplied by 6.5 equal units of measure the radius of the circle = 20.42. 20.42 multiplied by the radius of the circle 6.5 equal units of measure = 132.73.

So the area of the circle is 132.73.

The width of the square is 11.52 equal units of measure.

11.52 multiplied by 11.52 or 11.52 squared = 132.7104.

So if the results are reduced to 4 decimal places then the area of the circle and the area of the square can be the same.

The width of the square is 11.52 equal units of measure and 11.52 multiplied by 11.52 = 132.7104. 11.52 squared = 132.7104. So the area of the circle is close to the width of the square according to the modern version of Pi approximated to 3.141538461538462. The diameter of the circle with 13 equal units of measure divided by the width of the square with 11.52 equal units of measure produces the irrational ratio of 1.128472222222222. The radius of the circle with area of 132.7104 square units of measure again is 6.5 equal units of measure so if a rectangle is created with the longest length of the rectangle being 6.5 equal units of measure while the shortest length of the rectangle is 5.76 equal units of measure the diagonal of the rectangle will measure 48.45 degrees when the longest length of the rectangle is vertical. Regarding the creation of a rectangle with the longest length measuring 6.5 equal units of measure while the shortest length measures 5.76 units of measure the diagonal of the rectangle will measure 41.55 degrees when the longest length of the rectangle is horizontal and the shortest length of the rectangle is vertical. Knowing the angle that is formed between the 2 poles of the circle’s diameter and the centre edge of the square that is parallel to the diameter of the circle can make the creation of a circle and square with approximate equal areas quicker and easier.

If a scalene triangle is created with the second longest length being 6.5 equal units of measure while the shortest length of the scalene triangle is 5.76 equal units of measure then the longest length of the scalene triangle also called the hypotenuse is 8.68 equal units of measure. The ratio of a scalene triangle’s hypotenuse that measures 8.68 equal units of measure divided by the shortest length of the scalene triangle 5.76 equal units of measure is 1.506944444444444

So if a square and a circle have approximate equal areas then the measure for the diameter divided by the width of the square can produce the ratios 1.128472222222222 or 1.129 plus an infinite amount of decimal places or 1.125 if the diameter of the circle is 9 and the width of the square is 8. 1.130434782608696 is the irrational ratio that can be obtained when the diameter of a circle has 26 equal units of measure and is divided by the width of a square with 23 equal units of measure. 26 divided by 23 = 1.130434782608696.

To make that above statement again clearer if a circle and a square are created with approximate equal areas of measure then when the measure for the diameter of the circle is divided by the measure for the width of the square then the result will be any of the following ratios depending on the amount of accuracy that is used:

1.13.

1.128 plus an infinite amount of decimal places.

1.129 plus an infinite amount of decimal places.

1.130 plus an infinite amount of decimal places.

1.125 when the diameter of the circle is 9 or a multiple of 9 and the width of the square is 8 or a multiple of 8.

“Creating a circle and a square with approximate equal areas from whole numbers”:

To obtain the area of a circle any version of Pi must be multiplied by the radius of the circle and the result of Pi being multiplied by the radius of the circle must then be multiplied by the radius of the circle for the measure of the circle’s area to be known. So in this case that is 22 divided by 7 this approximation version of Pi = 3.142857142857143 multiplied by 4.5 = 14.142857142857143 multiplied by 4.5. = 63.642857142857143. So the area of a circle with a diameter of 9 equal units of measure according to the ancient approximation version of Pi 22 divided by 7 = 3.142857142857143 is 63.642857142857143.

The area of a square can be known by multiplying the width of the square by the width of the square. So in this case the width of the square has 8 equal units of measure and 8 multiplied by 8 = 64. 8 squared = 64. So the area of a square with a width of 8 equal units of measure is 64 equal units of measure. So the area of the circle is 63.642857142857143 equal units of measure, while the area f the square is 64 equal units of measure, when a square and a circle is created to have equal areas and the diameter of the circle has 9 equal units of measure while the width of the square has 8 equal units of measure. Both the circle and the square must share the same centre. So the area of the circle is 0.36 % away from the measure for the area of the square according to the approximation version of the modern version of Pi that = 3.142857142857143. So when the values of the areas of both the circle and square are read without an infinite amount of decimal places for the area of the circle the area of the circle read 63.64 equal units of measure while the area of the square reads 64 equal units of measure.

Again the width of the square has 8 equal units of measure and 8 multiplied by 8 = 64. 8 squared = 64. So the area of a square with a width of 8 equal units of measure is 64 equal units of measure and if the radius of the circle has 4.5 equal units of measure the diameter of the circle has 9 equal units of measure. The circumference of a circle with a diameter of 9 equal units of measure is 28.27 units of measure. So Pi can be approximated to 3.141111111111111. If Pi approximated to 3.141111111111111 is multiplied by the radius of the circle the result is 14.135 and 14.135 multiplied by 4.5 the radius of circle says that the area of a circle with a diameter of 9 equal units of measure is 63.6075.

If a scalene triangle is created with the second shortest length being equal to 13 equal units of measure while the shortest length is equal to 11.52 equal units of measure then the longest length is the hypotenuse and that measures 17.36 equal units of measure. 17.36 divided by 11.52 also equals 1.506944444444444.

“Another version for creating a circle and a square with approximate equal areas”.

The following version for creating a circle and a square with approximate equal areas involves the diameter of the circle having 7 equal units of measure while the width of the square has 6.2 equal units of measure plus the 22 divided by 7 approximation of modern Pi. 22 dived by 7 = 3.142857142857143.

So in this case 22 divided by 7 = 3.142857142857143 multiplied by 3.5 the radius of the circle = 11 multiplied by 3.5 the radius of the circle = 38.5.

The width of the square is 6.2 equal units of measure. 6.2 squared = 38.44 equal units of measure. Also 6.2 multiplied by 6.2 = 38.44 equal units of measure. So again the width of the square is 6.2 equal units of measure and a square with a width of 6.2 equal units of measure has an area of 38.44 units of measure. The radius of the circle is 3.5 equal units of measure and a circle with a diameter of 7 equal units of measure has a radius of 3.5 equal units of measure and again a circle with a radius of 3.5 equal units of measure has an area of 38.5 equal units of measure according to the approximation value of modern Pi that is 22 divided by 7 = 3.142857142857143. Regarding the third version for the creation of a circle and a square with equal areas the area of the circle is 0.06 larger than the area of the square and that means that the measure for both the area of the circle and the square are very close.

“Creating a Circle and square with equal areas from existing measurements of a circle and a square with equal perimeters”:

If a circle with a circumference equal to the perimeter of a square has already been created and the new desire is to create a square with an area approximately equal to the area of the existing circle that has a circumference equal to the perimeter of another square then a solution is to obtain the ratio of the diameter of the circle divided by the edge length of the square with a perimeter equal to the circumference of the circle with the appropriate diameter and then apply the result of the circle’s diameter being divided by the edge of the square with a perimeter equal to the circumference of the circle to square root and the result is the ratio of the diameter of the existing circle divided by the edge length of a square with an area approximately equal to the area of the existing circle.

An Example follows:

1. Measurements of a circle and square with equal perimeters derived from Pi approximated to 3.141361256544503: Diameter of circle is 11.46 equal units of measure and edge length of square is 9 equal units of measure. Perimeter of square is 36 and circumference of circle is 36.

The ratio of the diameter of the circle divided by the edge length of the square is 1.273333333333333 and can also be obtained if 4 is divided by Pi approximated to 3.141361256544503 = 1.273333333333333.

2. If the ratio 1.273333333333333 is applied to square root meaning the square root of 1.273333333333333 = 1.128420725320717.

3. The ratio 1.128420725320717 can tell us the measure for the correct edge length of the square with an area approximately equal to the area of a circle with a diameter of 11.46 equal units of measure. A square with an edge length of 10.155786527886455 equal units of measure is approximately equal to the area of a circle with a diameter that has 11.46 equal units of measure.

4. 10.155786527886455 multiplied by 10.155786527886455 = 103.14. 10.155786527886455 squared = 103.14. Pi approximated to 3.141361256544503 multiplied y 5.73 equal units of the measure being the radius of the circle = 18 and 18 multiplied by 5.73 = 103.14. 103.14 is the area of a circle with a diameter of 11.46 equal units of measure according to Pi approximated to 3.141361256544503. Remember that the measure for the edge of the square must be reduced to 4 decimal places so that means that it is usually impossible for the area of the square and the area of the circle to have exactly the same measure of square units. The edge length of the square reduced to 4 decimal places is 10.15 and 10.15 squared is 103.225. So both the area of the square and the circle are quite close but not exactly the same in measure.

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