Cut a honeycomb across and every cell is a hexagon, the same shape repeated with no gaps between the cells and no waste. Draw lines from the centre of one cell to each of its six corners and the hexagon falls apart into six equilateral triangles, all identical.
That is why the shape works. Six equilateral triangles fit around a point exactly, because each contributes sixty degrees and six sixties make a full turn. Nothing is left over and nothing overlaps.
Squares also tile a surface without gaps, and so do triangles. What sets the hexagon apart is the amount of wall required. For a given area, a hexagonal grid needs less wall than a square grid, and much less than a triangular one — and wall means wax, which the bees must make from honey.
Mathematicians suspected for a very long time that the hexagon is the best possible answer to this problem, and a complete proof arrived only in modern times. The bees did not wait for it.
There is a second reason the shape suits a hive. A hexagon is close to a circle, which is the most efficient shape of all — but circles leave gaps between them, and a hexagon does not.
The useful part for a student is smaller and closer to hand. Any question about a regular hexagon can be turned into a question about an equilateral triangle, and any question about an equilateral triangle becomes two right-angled triangles the moment you drop a line from its apex.