It always adds up to the number of sides!

That's fascinating.  I don't think I have the time to prove it right now, though.  That being said, I think my algorithm is working because it was discovering this pattern :)

I also don't know how you go about decomposing

$$a^2 + b^2 + c^2 + \ldots = n$$

into the individual length components.

SimpleStacker

science

[Daily puzzle] Find a little known property of regular polygones

south_korea_ln

Hint: 

If one defines the lengths of the colored segments with letters a, b, c, d, e, etc...
What do they add up to when taking the sum of their squares?

Example, for the last one (with 3 segments):
$$
a^2 + b^2 + c^2 = ?
$$

Do this for each polygon and discover a pattern...