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Pi as a circle's circumference integrated using two points and a right triangle

ewmon
in Analysis and Calculus

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I got to thinking about this from two recent threads on the nature of pi.

 

We're all familiar with scribing a semi-circle using two points (A and C) and a right triangle, shown below.

 

001-planegeom-right-triangle-inscribed-c

 

How then would the distance along the circumference of the semi-circle be computed using integration? Can it be done without using trigonometry, or is trig inherently part of the process?

 

Pi would then be the ratio of the semi-circle's circumference to ½ of AC.

 

I apologize for not thinking this out myself, but I haven't focused in this mathematical direction in quite some time.

  • 1 month later...

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