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
=
.
Solution Stats
Solution Comments
Show commentsProblem Recent Solvers5
Suggested Problems
-
126 Solvers
-
71 Solvers
-
Right Triangle Side Lengths (Inspired by Project Euler Problem 39)
1738 Solvers
-
170 Solvers
-
Cumulative maximum of an array
34 Solvers
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!