Can Geometry Alone Provide an Accurate Estimation of Pi?

[kalikusu]

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Have neither seen an estimation nor derivation of pi that does not use trig functions. This is problematic as trig functions require radian inputs, via the relation pi radians = 180 deg. But if looking for pi, then how to get the input for the trig functions without pi?

Sure there are calculators that allow degree inputs provided, the calculator is set for DEG mode. Internally, the degrees are converted to radians before the trig function is evaluated.

Using Geometry alone, namely equation for a circle, the tangent to any point on the circle intersecting with a normal line to determine the length of an inscribed or circumscribed polygon can be used but with problems.

If focus on polygons starting with 4 sides and doubling with each iteration starting with the normal y = x, then the angle between a chosen segment of the polygon will halve with each iteration. On a spreadsheet, the estimate for pi then approaches 4.

If start with the normal y = x to identify the mid-point of the chord for the inscribed polygon and the point on the tangent to the circumscribed polygon, then the next normal has to be determined via the tangent to that point. But if do this, then after the 15th iteration , the estimate for pi becomes erratic.

 

[berkeman]

Welcome to PF.

Per the PF rules (see INFO at the top of the page) and per the stickie thread at the top of this forum, the New Member Introduction forum is not for questions, so this thread is locked.

Please re-post your question in the General Math forum, but with some improvements:

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** Please always include links to the reading you have been doing about your question. If your best estimate of $\pi$ is 4.0, then obviously you have not been reading the right introductory references...