Area +3 to completely cover a quarter circle with integral area and slider

This is the value of pi based on 4/√φ.

Note that for any radius r greater than 2,600 that the UPPER LIMIT for pi is LESS than 4/√φ, which proves it to be a false value for pi that overstates the true value of pi. Why 2,600? That's because the difference between traditional pi and 4/√φ is a little less than 1 in 1,000, so it takes a level of precision slightly more than that to reveal the truth about their differences, and this version adds an extra square on the grid for more margin of proof yet.

This proof does not depend on lines of polygons that approach the circumference of the circle, nor straight lines that must approximate curved lines.To the contrary, it intentionally overstates the area of the circle by identifying the squares on the grid that it takes to completely paint over and cover the entire circle. It thus overstates the value of pi, but in doing so identifies the upper limit that must exceed the true value of pi. This proof is also not violated by the principle of isoperimetric inequality, as it does not attempt to create an area equal to that of the circle, but rather to intentionally exceed it for the purpose of determining whether π=4/√φ can pass the test of being the true value for pi, which it does not. A boundary with an even greater coverage of the circle could be expressed in a formula which would only further validate this truth.

Rather than comparing the calculated value for Pi in this test, another alternative is to compare the areas of the upper limit area to that of a circle with Pi=3.14159... (which is within the upper limit) or PI=4/√φ (which EXCEEDS the upper limit.)

Area of a circle based on Pi=3.14159...:

Area of the UPPER LIMIT:

Area of a circle based on Pi=4/√φ: