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. ).
  • 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 = .

Solution Stats

80.0% Correct | 20.0% Incorrect
Last Solution submitted on Aug 04, 2023

Solution Comments

Show comments

Problem Recent Solvers4

Suggested Problems

More from this Author116

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!