The sum of the internal angles of a polygon with n sides is 180°·(n -2). Then the internal vertex angle will have measure ...
... internal angle = 180°·(n -2)/n
In order for the polygon to tile a plane, that angle must be a submultiple of 360°. That is, ...
... 360°/(internal angle)
must be an integer. Or, ...
... 360°/(180°(n -2)/n) = 2n/(n -2) . . . must be an integer
This is the case for n ∈ {3, 4, 6}, a regular triangle, square, and hexagon.
