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Converging Circumference (Posted on 2004-08-09) Difficulty: 5 of 5
Draw a unit circle.
Around it, circumscribe an equilateral triangle.
Circumscribe another circle around that.
Circumscribe a square around this circle.
Circumscribe another circle around that.
Circumscribe a regular pentagon around this circle.
Circumscribe another circle around that.

Continue, ad infinitum, with the next regular polygon.

Do the radii of these circles converge? If so, what is the limiting radius?

No Solution Yet Submitted by ThoughtProvoker    
Rating: 2.7500 (4 votes)

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Some Thoughts first steps | Comment 1 of 4

If an N-gon has a circle inscribe in it and a circle circumscribed about it, the circle's sizes are proportional.  If the smaller circle is a unite circle, then the larger circle has a radius of csc(90-180/N).

Therefore, the limiting radius (if there is one) is the infinite product of csc(90-180/N) for all integers N greater than 2.

I wrote a short calculator program to do this equation over and over again, and it's still going.  It's going extremely slowly at 8.67, but doesn't seem to be stopping.  Chances are the program will stop before I get a more definite answer.

Edit:  Well, obviously, csc(90-180/N) is the same as sec(180/N).  I only realized that after seeing Charlie's comment.

Edited on August 9, 2004, 1:37 pm
  Posted by Tristan on 2004-08-09 12:54:49

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