What regular polygons can be used to make a regular tesselations?

Answer 1

A regular polygon's angles are measured by the formula 180 * (n - 2) / n.

Regular polygons will only tesselate if m * ( 180 * (n - 2) / n ) = 360, where m is an integer.

Let's go through all the possible regular polygons.

3 sided polygon: m * ( 180 * 1 / 3 ) = 360 -> 60m=360 -> m=6, Able to tesselate

4 sided polygon: m * ( 180 * 2 / 4 ) = 360 -> 90m=360 -> m=4, Able to tesselate

5 sided polygon: m * ( 180 * 3 / 5 ) = 360 -> 108m=360. Not able to tesselate

6 sided: m * ( 180 * 4 / 6 ) = 360 -> 120m=360. Able to tesselate

We do not need to check more, for the polygons that are able to tesselate have a decreasing m value, from 6 to 4 to 3. The next possible m value would be 2, and we know this cannot happen, because if m = 2, then the polygon would have to have angles of 180 degrees; impossible.

Therefore, we can only tesselate using triangles, squares, and hexagons.