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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)

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Solution and proof Comment 4 of 4 |
Let X_1 be the polygon with the greatest area, with L being the length of the divergent side. 

Construct a congruent polygon X_2. Lay X_1 and X_2 beside each other so that the divergent sides (both of length L) are aligned. This gives a 4044 sided regular polygon. 

If follows that the area of X_1 is one half the area of a 4044 sided regular polygon. 

That was the insight. Now for the calculation:

Let Y be a 4044 sided regular polygon with side length 1. Let Z be the circle circumscribing Y. Let W be the triangle with vertices at U (= the centre of Z) and any two adjacent points of the polygon. The angle W measured at U is v = 2 pi / 4044. Then sin( v/2 ) = (1/2) / r, where r is the radius of the circle. So r = 1/2 csc (v/2), and the area of W is given by 

K_T = (1/2) * 1 * r cos (v/2) = (1/4) cot (v/2).

The area of the polygon is one half the area of 4044 such triangles

K_P = (1/2) 4044 (1/4) cot (pi / 4044) = (1011 / 2) cot (pi / 4044)
       = 505.5 * (1287.245) = 650702 (near enough)

  Posted by FrankM on 2023-05-31 13:42:04
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