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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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Solution Web solution | Comment 3 of 4 |
Using basic trigonometry I found that the radio grwos in proportion to 1/cos(π/n) when you add the n-sided polygon.


If we start with a unic circle then the limiting radius is the inverse of the product cos(π/3)xcos(π/4)xcos(π/5)x...


I couldn't calculate this so turned to the web and found an interesitng site at http://icl.pku.edu.cn/yujs/MathWorld/math/p/p456.htm where it's shown that the limit exists and is finite and about 8.7000366...
  Posted by Oskar on 2004-08-09 14:21:32

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