What do bathroom tiles, honeycombs, and turtle shells have in common? They're all covered by a single repeating shape that fills the plane with no gaps and no overlaps. This kind of regular arrangement is called "tessellation." Remarkably, though, only three regular polygons — the equilateral triangle, the square, and the regular hexagon — can tile a plane using that shape alone. No matter how hard you try, a regular pentagon or heptagon always leaves gaps.
The reason lies hidden in the angles. For several copies of a shape to meet at a point and fill the plane, the angles gathered around that point must add up to exactly 360°. An equilateral triangle's interior angle is 60°, and 60°×6=360°, so six triangles meeting at a point fit perfectly. A square's interior angle is 90°, so 90°×4=360° — four of them fill the space perfectly. A regular hexagon's interior angle is 120°, so 120°×3=360° — three of them fit exactly. These are the only three cases where a whole number of copies divides 360° evenly, which is why, among the regular polygons, only these three can tile a plane "on their own."
Why doesn't the regular pentagon work? Its interior angle is 108°. Dividing 360 by 108 gives about 3.33, which isn't a whole number. Gather 3 of them and you get 324°, leaving a 36° gap; gather 4 and you get 432°, which actually overlaps by 72°. So a regular pentagon can never tile a plane on its own. In fact, if you look closely at a soccer ball, it's a mix of pentagons and hexagons — pentagons alone can't cover a plane (or a sphere), so they need hexagons alongside them to complete the ball's shape.
It's no accident that bees build their honeycombs as hexagons either. It's been mathematically proven that, among the equilateral triangle, square, and regular hexagon, the hexagon encloses the largest area for the same amount of material (the same perimeter). Bees have essentially evolved to naturally pick the most efficient shape — the hexagon — so they can store the most honey using the least wax. The artist M.C. Escher is famous for using this tessellation principle to create mesmerizing works where complex shapes like birds, fish, and lizards interlock without a single gap.
When studying this with kids, the surest method is to cut out several paper copies of a regular polygon and let them lay the pieces out on a flat surface themselves. Feeling with their own hands that triangle, square, and hexagon pieces fit together with no gaps, while pentagon or heptagon pieces always leave gaps or overlap no matter how you rotate them, builds a much surer understanding of "why only these three work" than any formula. It's also a great extension activity to look for cases where a single regular polygon can't tile a plane alone, but mixing two or more shapes (like squares and regular octagons) can.
On our activity page, you can pick a regular polygon and gather several copies at a single vertex to see with your own eyes whether the angles add up to exactly 360°. If you pick a shape that doesn't fit perfectly, the leftover gap angle is even shown, so you can compare for yourself why the equilateral triangle, square, and regular hexagon are so special.