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.
