All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math > Calculus
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts possible approach | Comment 3 of 4 |

After a few false starts...

The formula for the number of diagonals is n(n-3)/2, as is confirmed by OEIS A000096

Starting with, say, a regular undecagon in the unit circle, the different length values including the sides are 0.563465..., 1.081282..., 1.51150098..., 1.81926620..., and 1.9796452... However, the side value is not needed for current purposes, leaving 4 diagonals for each of the 11 vertices for a total of 44 in all. 11*8/2=44, as expected.

Those values correspond to 2*sin(pi/11) - for the unused side - 2*sin(2pi/11), 2*sin(3pi/11), 2*sin(4pi/11), and 2*sin(5pi/11). Checking: 0.563465114..., 1.081281635..., 1.511499149..., 1.819263991...,1.979642884....

So for the total length of the diagonals at a single vertex of any 'oddagon', we have:

sum for k = 2 to (n-1)/2 {2sin(pi*k/n)} where n is the number of vertices. Call the sum, S.

Then we need to multiply S by the number of vertices, n, and divide by n(n-3)/2 to obtain the mean length.

The mean is therefore 2S/(n-3). Examples:

for n=2 to 44 sum {2sin(pi*n/89)} Mean = 1.315876594
for n=2 to 90 sum {2sin(pi*n/181)} Mean = 1.294276112
for n=2 to 498 sum {2sin(pi*n/997)} Mean = 1.277068583
for n=2 to 8000 sum {2sin(pi*n/16001)} Mean = 1.273478254

for n=2 to 50000 sum {2sin(pi*n/100001)} Mean = 1.273277741

and the sums appear to converge nicely on 4/pi. Reassuringly, the same result occurs if the two points are picked at random on the circumference:, and see 

Edited on November 24, 2018, 7:16 am
  Posted by broll on 2018-11-24 06:44:42

Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2022 by Animus Pactum Consulting. All rights reserved. Privacy Information