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Maximum Polygon Area Ascertainment (Posted on 2023-05-30) Difficulty: 3 of 5
• P is a convex, near regular 2023-sided polygon.
• Precisely 2022 of its sides have length 1, but the remaining side has a length different from 1.

Determine the maximum area of the polygon.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Calculating the intuition; not a proof | Comment 2 of 4 |
Probably a half circle; so the diameter would be the 2023rd side should be 4044/pi
More accurately, figure 1/2 the area of a totally regular polygon of 4044 sides

Regular poly of n equal sides and side length s:
Area =  n*s^2 /(4* tan(pi/n))

Area of 4044-gon with s=1:  1301404.61490485
Half of that is 650702.307452425  (assumed maximum area for the 2023 sided polygon)

The "diameter" is twice the "radius":  sin(pi/n) = (s/2)/radius
radius = (s/2)/sin(pi/n)
radius = s/(2*sin(pi/n))
2*radius = 2023rd side length = 1/(sin(pi/n)) = 1287.24530920272 ****

which is pretty close to 4044/pi = 1287.24517972725

But this is not a proof or an optimization

  Posted by Larry on 2023-05-30 13:46:07
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