Geometric Proofs of Pi

   Welcome to the Geometric Proofs of Pi section of our Measuring Pi Squaring Phi web site.

  This section is divided into two parts:  (1) A Geometric picture of each Proof including an accompanying step by step summary of proof, and (2) A video of my stepping the reader through each step of each proof.  The best way to view this section is to first do Part (1) View the Geometric Proof Photo, and then walk through Part (2), its video explanation.  These geometric proofs are shown in chronological order of development as I learned more and more about how to discover the true value of Pi.

 

Geometric Proof 1 for True Value of Pi:

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Geometric Proof 1 for True Pi

 

 

 

 

 

 

 

 

 

 

Walk-through for Geometric Proof 1 for the True Value of Pi

Note:  A few readers have asked, “How do you know that the diameters of 4 blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 1 Fig 1?”

 Answer:

Blue circle diameter = 2 / Pi, given where Circumf = 2, and diam = C / Pi, Proof 1 Fig 1,

Big Yellow circle diameter = 2 sqrt Phi, from Kepler’s Triangle Proof 1 Fig 1.

 In Proof 1 Fig 1, we do not know yet what Pi is and so we ask, What is the value of Pi when the diameter of the Big Yellow Circle = 4 times the diameter of the Blue circle?  The answer is:

    2 sqrt Phi = 4 (2 / Pi),   Solve for Pi

   Pi = 4 / sqrt Phi,

   Pi = 4 / 1.272019650… = 3.144605511…

   Therefore, the diameter of 4 Blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 1 Fig 1 when Pi = 3.144605511… .  And since Pi is a universal constant, not a variable, there is no need to look for another value of Pi.

 Also, we can square the circumference of the Big Yellow Circle to the perimeter of Square YHWT, equate the two equations for C = P, and then solve for the value of Pi:

    P, Perimeter of Square YHWT = 8, given in Proof 1 Fig 1,

   C, Circumference of Big Yellow Circle = 2 Pi (sqrt Phi), from Kepler’s Triangle, Proof 1 Fig 1.

 When C = P, what is the value of Pi?

 Answer:

    C = 2 Pi (sqrt Phi) = P = 8,   Solve for Pi

   Pi = 8 / (2 sqrt Phi),

   Pi = 4 / sqrt Phi = 4 / 1.272019650… = 3.144605511…

    Therefore, we have squared the circumference, C, to the perimeter, P, and equated C = P to solve for the only variable left, which is Pi = 4 / sqrt Phi.  All of the above is clearly shown in Proof 1 Fig 1:  (a) the Big Yellow Circle’s diameter is 4 times the diameter of the Blue circle, and (b) we have squared the Big Yellow Circle’s circumference to the perimeter of Square YHWT using the construction and sides of Kepler’s Golden Ratio Right Triangle.

    Thanks for your questions.  Time for you to step through the video of Proof 1 Fig 1:

 

 

Geometric Proof 2 for True Value of Pi:

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Geometric Proof 2 for True Pi

 

 

 

 

 

Walk-through for Geometric Proof 2 for the True Value of Pi

Note:  A few readers have asked, “How do you know that the diameters of 4 blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 2 Fig 2?”

 Answer:

 Blue circle diameter = Pi, given where Circumf = C = diam x Pi = Pi^2, Proof 2 Fig 2,

Big Yellow Circle diameter = 2 sqrt Phi (Pi^2 / 2), from Kepler’s Triangle Proof 2 Fig 2.

 In Proof 2 Fig 2, we do not know yet what Pi is – except that we really do know from Proof 1 Fig 1, but let’s continue with Proof 2 Fig 2 as if we didn’t know — and so we ask, What is the value of Pi when the diameter of the Big Yellow Circle = 4 times the diameter of the Blue circle?  The answer is:

    2 sqrt Phi (Pi^2 / 2) = 4 Pi,   Solve for Pi

   Sqrt Phi (Pi) = 4 ,

   Pi = 4 / 1.272019650… = 3.144605511…

    Therefore, the diameter of 4 Blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 2 Fig 2 when Pi = 3.144605511… .  And since Pi is a universal constant, not a variable, there is no need to look for another value of Pi.

 Also, we can square the circumference of the Big Yellow Circle to the perimeter of Square YHWT, equate the two equations for C = P, and then solve for the value of Pi:

    P, Perimeter of Square YHWT = 4 Pi^2, given in Proof 2 Fig 2,

   C, Circumference of Big Yellow Circle = 2 Pi (sqrt Phi) (Pi^2 / 2), from Kepler’s Triangle, Proof 2 Fig 2.

 When C = P, what is the value of Pi?

 Answer:

    C = 2 Pi (sqrt Phi) (Pi^2 / 2) = P = 4 Pi^2,   Solve for Pi

   Pi (sqrt Phi) = 4,

   Pi = 4 / sqrt Phi = 4 / 1.272019650… = 3.144605511…

    Therefore, we have squared the circumference, C, to the perimeter, P, and equated C = P to solve for the only variable left, which is Pi = 4 / sqrt Phi.  All of the above is clearly shown in Proof 2 Fig 2:  (a) the Big Yellow Circle’s diameter is 4 times the diameter of the Blue circle, and (b) we have squared the Big Yellow Circle’s circumference to the perimeter of Square YHWT using the construction and sides of Kepler’s Golden Ratio Right Triangle.

    Thanks for your questions.  Time for you to step through the video of Proof 2 Fig 2:

 

Geometric Proof 4 for True Value of Pi:

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Geometric Proof 4 for True Pi

 

Walk-through for Geometric Proof 4 for the True Value of Pi

 

 

Geometric Proof 6 for True Value of Pi:

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Geometric Proof 6 for True Pi

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Walk-through for Geometric Proof 6 for the True Value of Pi