@Quehoniaomath,
To get the correct measure for a circle’s diameter and to prove that Golden Pi = 3.144605511029693 is the true value of Pi by applying the Pythagorean theorem to all the edges of a Kepler right triangle when using the second longest edge length of a Kepler right triangle as the diameter of a circle then the shortest edge length of a Kepler right triangle is equal in measure to 1 quarter of a circle’s circumference. Also if the radius of a circle is used as the second longest edge length of a Kepler right triangle then the shortest edge length of a Kepler right triangle is equal to one 8th of a circle’s circumference:
Example 1:
The circumference of the circle is 12 but the measure for the diameter of the circle is not yet known. The second longest edge length of a Kepler right triangle is used as the diameter of a circle in this example. 12 divided by 4 is 3 so the shortest edge length of the Kepler right triangle is 3. The hypotenuse of a Kepler right triangle divided by the shortest edge length of a Kepler right triangle produces the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989.
According to the Pythagorean theorem the hypotenuse of any right triangle contains the sum of both the squares on the 2 other edges of the right triangle.
So the shortest edge length of the Kepler right triangle is 3 and since the ratio gained from dividing the hypotenuse of a Kepler right triangle by the measure for the shortest edge of the Kepler right triangle is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989 then the measure for the hypotenuse of a Kepler right triangle that has its shortest edge length as 3 is 4.85410196624967. 4.85410196624967 divided by 3 is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989.
4.85410196624967 squared is 23.562305898748912.
3 squared is 9.
23.562305898748912 subtract 9 = 14.562305898748912.
The square root of 14.562305898748912 is 3.816058948542188.
Remember that the second longest edge length of the Kepler right triangle is used as the diameter of a circle. So the measure for both the second longest edge length of this Kepler right triangle and the diameter of the circle is 3.816058948542188. Remember that the shortest edge length of this Kepler right triangle is 3 and is equal to 1 quarter of a circle’s circumference that has a measure of 12 equal units.
Circumference of circle is 12
Diameter of circle is 3.816058948542188.
12 divided by 3.816058948542188 = 3.144605511029709. So Golden Pi = 3.144605511029709 is the true value of Pi.
Example 2:
The circumference of the circle is 45623096 but the measure for the diameter of the circle is not yet known. The second longest edge length of a Kepler right triangle is used as the radius of a circle in this example. 45623096 divided by 8 is 5702887 so the shortest edge length of the Kepler right triangle is 5702887. The hypotenuse of a Kepler right triangle divided by the shortest edge length of a Kepler right triangle produces the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989.
According to the Pythagorean theorem the hypotenuse of any right triangle contains the sum of both the squares on the 2 other edges of the right triangle.
So the shortest edge length of the Kepler right triangle is 5702887 and since the ratio gained from dividing the hypotenuse of a Kepler right triangle by the measure for the shortest edge of the Kepler right triangle is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989 then the measure for the hypotenuse of a Kepler right triangle that has its shortest edge length as 5702887 is 9227465. 9227465 divided by 5702887 is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989.
9227465 squared is 85146110326225.
5702887 squared is 32522920134769.
85146110326225 subtract 32522920134769 = 52623190191456.
The square root of 52623190191456 = 7254184.32295844.
Remember that the second longest edge length of the Kepler right triangle is used as the radius of a circle. So the measure for both the second longest edge length of this Kepler right triangle and the radius of the circle is 7254184.32295844. Remember that the shortest edge length of this Kepler right triangle is 5702887 and is equal to one 8th of a circle’s circumference that has a measure of 45623096 equal units.
Circumference of circle is 45623096.
Radius of circle is 7254184.32295844.
Radius of circle is 7254184.32295844 and multiplied by 2 = 14508368.64591688.
Diameter of circle = 14508368.64591688.
45623096 divided by 14508368.64591688 = 3.14460551102965. So Golden Pi = 3.14460551102965 is the true value of Pi.
Example 3:
The circumference of the circle is 8 but the measure for the diameter of the circle is not yet known. The second longest edge length of a Kepler right triangle is used as the radius of a circle in this example. 8 divided by 8 is 1 so the shortest edge length of the Kepler right triangle is 1. The hypotenuse of a Kepler right triangle divided by the shortest edge length of a Kepler right triangle produces the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989.
According to the Pythagorean theorem the hypotenuse of any right triangle contains the sum of both the squares on the 2 other edges of the right triangle.
So the shortest edge length of the Kepler right triangle is 1 and since the ratio gained from dividing the hypotenuse of a Kepler right triangle by the measure for the shortest edge of the Kepler right triangle is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989 then the measure for the hypotenuse of a Kepler right triangle that has its shortest edge length as 1 is 1.61803398874989. 1.61803398874989 divided by 1 is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989.
1.61803398874989 squared is 2.61803398874989.
1 squared is 1.
2.61803398874989 subtract 1 = 1.61803398874989.
The square root of 1.61803398874989 = 1.27201964951406.
Remember that the second longest edge length of the Kepler right triangle is used as the radius of a circle. So the measure for both the second longest edge length of this Kepler right triangle and the radius of the circle is 1.27201964951406. Remember that the shortest edge length of this Kepler right triangle is 1 and is equal to one 8th of a circle’s circumference that has a measure of 8 equal units.
Circumference of circle is 8.
Radius of circle is 1.27201964951406.
Radius of circle is 1.27201964951406 and multiplied by 2 = 2.54403929902812.
Diameter of circle = 2.54403929902812.
8 divided by 2.54403929902812 = 3.144605511029715. So Golden Pi = 3.144605511029715 is the true value of Pi.
Example 4:
The edge of the square is the second longest edge length of a Kepler right triangle.
The diameter of the circle is the hypotenuse of a Kepler right triangle.
The width and edge of the square in this example is 12 so both the perimeter of the square and the circumference of the circle have a measure of 48. 12 multiplied by 4 = 48. If the second longest edge length of a Kepler right triangle is divided by the shortest edge length of the Kepler right triangle then the result is the square root of the Golden ratio = 1.272019649514069. If the hypotenuse of a Kepler right triangle is divided by the shortest edge length of a Kepler right triangle the result is the Golden ratio of Cosine (36) multiplied by 2 = 1.61803398874989. The edge of the square is the second longest edge length of a Kepler right triangle and the width and edge of the square is 12. The second longest edge length of this Kepler right triangle is known to be 12 but the measure for both the shortest edge length of this Kepler right triangle and also the hypotenuse of this Kepler right triangle are not yet known. The measure for the shortest edge length of the Kepler right triangle can be known if the second longest edge length of the Kepler right triangle is divided by the square root of the Golden ratio = 1.272019649514069. The measure for both the width of the square and the second longest edge length of the Kepler right triangle is 12. 12 divided by the square root of the Golden ratio = 1.27201964951406 = 9.433816533089079. The shortest edge length of the Kepler right triangle in this example is 9.433816533089079. Apply the Pythagorean theorem to all the edges of this Kepler right triangle to discover the measure for the hypotenuse of a Kepler right triangle that has its second longest edge length as 12 while the shortest edge length of this Kepler right triangle is 9.433816533089079.
12 squared = 144.
9.433816533089079 squared = 88.996894379984853.
144 plus 88.996894379984853 = 232.996894379984853.
The square root of 232.996894379984853 = 15.26423579416883.
15.26423579416883 divided by 9.433816533089079 = The Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.
The hypotenuse of a Kepler right triangle that has its second longest edge length as 12 while the shortest edge length of this Kepler right triangle is equal to 9.433816533089079 is 15.26423579416883.
The diameter of a circle with a circumference of 48 is 15.26423579416883.
48 divided by 15.26423579416883 = Golden Pi = 3.144605511029693.