Regular polygons and tiling

A polygon with all its sides equal and all its angles equal is a regular polygon: an equilateral triangle, a square, a regular pentagon, a regular hexagon… Check both: a rhombus has equal sides but unequal angles, and a rectangle has equal angles but unequal sides, so neither is regular. As its sides are all equal, a regular polygon's perimeter (all its sides added up) = side × number of sides, and side = perimeter ÷ number of sides. Covering a flat surface with copies of one shape, no gaps and no overlaps, is tiling (tessellating). See whether the angles meeting at a point make exactly 360°: an equilateral triangle's angles are 60°, and 6 make 360°; a square's are 90°, and 4 make 360°; a regular hexagon's are 120°, and 3 make 360°. All three tile. A regular pentagon's angles are 108°: 3 make 324° and 4 make 432°, never 360°, so it can't tile. Another way: cut a few out of paper and try fitting them together, looking for gaps.