

Circle Ratio 3.075
Name: Byron
Status: educator
Grade: other
Location: AZ
Country: USA
Date: Fall 2012
Question:
If I divide a circle into sectors, then take one sector and divide it in half, and extend all the dividing rays well beyond the circle, I can then place a chord to intersect the circle where two of the rays have been divided in half. One inside the circle, and one in contact with the circle just out side of it, at a perfect right angle to it. And by measuring either the difference between the length of the two chords, or the length between the chords: as measured along the dividing ray, I get the same answer. I then divide that answer in two, and can either subtract it from the length of the longer chord, or add it to the length of the shorter chord. I also get the same answer for those two ultimate calculations. I then divide the number of degrees of the central angle (that I chose to divide the circle) into 360 degrees and multiply the answer times the arc circumference that I got from all of the above original computation.
No matter how many sectors (both even and odd) that I divide the circles into, (and I always try to be as accurate as I can be when measuring all the dimensions) every time I divide r into c I keep getting 3.075 exactly. No unending decimal or compound fraction to deal with.
I seem to recall that some well know mathematician did this procedure, as I saw it in a documentary about Archemedes but I do not know who it was, or what the documentary was from. Does it ring a bell? I would like to read or watch something like that again.
Replies:
You have an assumption that is which is fundamentally flawed. Any method that involves mechanical measurement – dividing a sector, matching a contact angle, etc. – involves experimental errors, no matter how sharp your pencil (pen) is, no matter how smooth your paper is, or how straight your ruler is. Determining that the ratio of the circumference, C, and the diameter, d – that is C/d = pi is the result of a mathematical proof, with no mechanical constructions allowed. You already know that you have a systematic error of about  3.7%. Also your experimental precision, 3.XXX, is 0.00X which is only about 6 parts in 3075. That does not even come close to “proving” the ratio of C/d. That ratio, by mathematical PROOF cannot be a rational number no matter how good your mechanical drawing is.
An addition to the answer above: See http://en.wikipedia.org/wiki/Pi that provides many interesting details about the irrational and transcendental nature of pi.
Vince Calder
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