If a polygon is a 2D (flat) figure made up of
line segments which are connected to form a closed chain, then
a regular polygon or n-gon
is a polygon of which all angles are the same and of which all sides have the same length. Some examples:
4-gon: a square
5-gon: a pentagon
6-gon: a hexagon
Etc.
For all n-gons, the following rules hold:
The inside angles are all:
in degrees: 180(n - 2) / n
in radians: π(n - 2) / n
The outside angles are all:
in degrees: 360 - inside angle
in radians: 2π / inside angle
As the number of angles (n) of an n-gon increases, the inside angles of the n-gon also increase.
However, the size of the inside angles does not increase linearly with the increase in the number of
angles. I.e., if the number of angles doubles; e.g., from a square to an octagon,
the size of the inside angles less than doubles (from 90° to 135°).
The graph below shows this.
So what is the largest possible inside angle of a regular polygon? Well, that would be the limit of
180(n - 2) / n with n approaching infinity:
\[ \lim_{n \to \infty} \frac{180(n - 2)}{n} = 180 \]
What sort of a polygon would that be?
Play with this below:
Number of sides (3 ≤ n ≤ 50):
Inside angle (degrees) with increasing number of sides