Question

What kinds of regular polygons can be used for regular tesselations

Answer 1

Answers are four, six and three sided

Answer 2

To tessellate a surface using a regular polygon, the interior angle must be a sub-multiple (i.e. factor) of 360 degrees to cover completely the surface.

For a regular three-sided polygon, the interior angle is (180-360/3)=60 °
       Since 6*60=360, so a regular three-sided polygon (equilateral triangle)            tessellates.
For a regular four-sided polygon, the interior angle is (180-360/4)=90 °
       Since 4*90=360, so a regular four-sided polygon (square) tessellates.
For a regular five-sided polygon, the interior angle is (180-360/5)=108 °
       Since 360/108=3.33... (not an integer), so a regular five-sided polygon           (pentagon) does NOT tessellate.
For a regular six-sided polygon, the interior angle is (180-360/6)=120 °
       Since 3*120=360, so a regular six-sided polygon (hexagon) tessellates.