In N-gons and their angles we explored regular n-gons; i.e., n-gons whose
angles and sides are all indentical.
We stated that the inside angles of those n-gons can be computed as follows:
Inside angles (in degrees) = 180(n - 2) / n
Although we only played with regular n-gons, that same inside angle rule applies to ALL n-gons, regular
or irregular.
We can prove this be dividing up any n-gon in triangles (aka
polygon triangulation. Since the sum of the (inside) angles of a triangle is 180°,
the total of all (inside) angles of an n-gon must be 180 × the number of triangles making up the polygon.
Since in a regular polygon all angles are equal, this implies that for a regular polygon all inside angles equal
the total of all triangle angles divided by the number of angles in the polygon. For example:
4-gon (square) can be divided into two triangles —> angle size = (2 × 180°) / 4 = 90°
5-gon (pentagon) can be divided into three triangles —> angle size = (3 × 180°) / 5 = 108°
6-gon (hexagon) can be divided into four triangles —> angle size = (4 × 180°) / 6 = 120°
We can play with this:
In the grid below, construct your own (simple) polygon by picking polygon points on the intersections of vertical and horizontal lines. To close (complete) a polygon, select the starting point.
Please note that a (simple) polygon is a closed figure (starting and end points are the same), of which the
line segments do not cross and do not 'double back' on themselves; i.e., angles of 180% between points are not allowed.