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Rounding the corners
OK, it is my turn to pose a problem --
A regular polygon having N sides is circumscribed on a circle having unit radius. [Note: all sides of a regular polygon are equal, and all of its angles are equal. 'Circumscribed' means that the midpoints of each of the polygon's sides are tangent to the circle.]
As a function of N, find an expression for the fraction of the polygon's area that lies outside the circle. In other words, what fraction of the polygon would have to be removed to leave the circle?
Added challenge (ala Monsieurcoffee's previous problem): show a derivation of your expression that does not make use of transcendental functions such as sine, cosine, tangent, etc.
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Richard Wallace
Mentor since 2011 for FRC 3620 Average Joes (St. Joseph, Michigan)
Mentor 2002-10 for FRC 931 Perpetual Chaos (St. Louis, Missouri)
 since 2003
I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.
(Cosmic Religion : With Other Opinions and Aphorisms (1931) by Albert Einstein, p. 97)
Last edited by Richard Wallace : 28-01-2003 at 21:21.
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