Problem 52844. Easy Sequences 42: Areas of Non-constructible Polygons

A constructible polygon is a regular polygon that can be constructed using only a compass and a straightedge.
Amazingly, Gauss found a way to identify which regular n-gon (abbreviation for a polygon, with n being the number of sides) is constructible, without even attempting to construct the polygon. Gauss's theorem states that an n-gon is contractible if and only if the totient of n is a power of 2. (The Euler Totient Function of a number n is defined as the number of integers from 1 to n that are coprime to n.)
For example, the 3-gon (equilateral triangle) is constructible because the totient of 3 is 2. Similarly, the 5-gon (regular pentagon) is constructible because the totient of 5 is . While, the 21-gon is non-constructible since the totient of is , not a power of 2.
For to , the number of sides of the n-gons that are constructible are as follows and their totients, , are all powers of 2. The non-constructible n-gons from 3 to are: , and their totients are .
Given the limit of the number of sides m, write a function that will output the sum of the areas of all non-constructible regular n-gons, for , inscribed in a unit circle (i.e. ).
NOTES:
  • Equality in float class is hard to establish. Therefore, for consistency, please round-off each area to 4 decimal places, before taking the total.
  • For , the function should return , because the sum of areas of regular polygons with sides = .
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75.0% Correct | 25.0% Incorrect
Last Solution submitted on Nov 30, 2025

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