How a hexagon closes the gaps
A circle encloses a lot of area for its boundary length, but identical circles cannot cover a flat surface by themselves: wherever three circles meet, small gaps remain. A hexagon solves the packing problem by sharing straight edges with its neighbors. Six sides let the tiles meet in a repeating pattern, so the surface is covered completely.
The key is that a regular hexagon’s angles are 120 degrees. Three corners meet at a point, and their angles add to 360 degrees—the full turn around that point. No gap or overlap is left behind. Equilateral triangles and squares can tile a plane for the same reason, but a regular hexagon’s outline is more circle-like than either.
Why that shape is efficient
For a fixed amount of boundary, a regular hexagon encloses more area than a square or an equilateral triangle. That makes it an efficient choice when a surface must be divided into equal-sized cells while keeping the total length of their walls relatively short. In a hexagonal tiling, each wall is shared by two cells, spreading its boundary cost between them.
This is the practical tradeoff behind the familiar comparison: circles are excellent at enclosing area in isolation, while hexagons come close to that efficiency and still fit together. A honeycomb’s cells can therefore be tightly packed without leaving empty spaces between them—a small geometric advantage repeated across the whole surface.
