Question

What is the range of possible values of internal angles of a regular `n`-sided polygon?

Answer 1

Essentially `[pi/3, pi)`, but only takes certain discrete values...

Explanation:

What is the range of possible values of internal angles of a regular `n`-sided polygon?

The sum of internal angles of a triangle is `pi` radians (`180^@`).

An `n`-sided polygon can be dissected into `n-2` triangles with vertices at the vertices of the polygon.

Hence each internal angle of a regular `n`-sided polygon is:

`(n-2)/n pi = (1-2/n)pi" "` radians

Note that `n >= 3`, and hence each internal angle is in the range:

`[pi/3, pi)`

The actual possible values are:

`pi/3, pi/2, (3pi)/5, (2pi)/3, (5pi)/7, (3pi)/4, (7pi)/9, (4pi)/5,...`