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Mean Diagonals (Posted on 2018-11-23) Difficulty: 2 of 5
Let Mn be the arithmetic mean of the lengths of the diagonals of regular n-gon with a circumradius of π(pi).

Compute limit of Mn as n tends to infinity.

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts one probably wrong way to go about it | Comment 1 of 4
The limit of such a regular n-gon is a circle with radius pi.

In polar coordinates if can be plotted as

r = 2*pi*cos(theta)

I'm tempted to say that the sought mean length would be

Integral{0 to pi}(2*pi*cos(theta) d(theta) / pi

but this does not take into consideration any clustering of the angles involved; it assumes the angles the diagonals make with tangents at a given point are uniformly distributed.  So I don't think it's worthwhile to use this integral.

  Posted by Charlie on 2018-11-23 11:42:02
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