Mathematics

Tessellation: Shapes That Tile Without Gaps

Shapes tessellate when the angles round each meeting point total exactly 360°. Why regular hexagons tile, regular pentagons cannot, and which three shapes do.

Tessellation

A tiling of a flat surface by shapes that fit together with no gaps and no overlaps, and which could continue for ever in every direction.

Also called: Tiling
Where students meet it: Grade 5 mathematics, alongside angles round a point, and again with interior angles of polygons at GCSE.

The short answer
A tessellation is a pattern of shapes covering a flat surface with no gaps and no overlaps. It works when the angles meeting at every corner add up to exactly 360°: three regular hexagons of 120° manage it, while regular pentagons of 108° cannot.

An example

Hexagon: 3 × 120° = 360° ✓ Pentagon: 3 × 108° = 324°, 4 × 108° = 432° ✗

Each interior angle of a regular hexagon is 120°, and three of them fit round a point exactly. A regular pentagon has interior angles of 108°: three leave a 36° gap and four overlap by 72°. No whole number of pentagons ever totals 360°, so regular pentagons cannot tessellate.

The 360° rule at every corner

Angles round a point add to 360°. In a tiling, every place where corners meet is such a point, so the angles gathering there must total exactly 360° — anything less leaves a gap and anything more forces an overlap.

That turns a question about shapes into a question about division. Take the interior angle of the shape and ask whether it divides into 360 a whole number of times. If it does, copies of the shape can close up round a point; if it does not, they cannot.

Only three regular polygons manage it alone

The equilateral triangle has 60° angles and six meet at a point. The square has 90° angles and four meet. The regular hexagon has 120° angles and three meet. That is the complete list, and no other regular polygon can be added to it.

Everything else fails the division test. The regular heptagon's angle is about 128.6°, which divides into 360 neither twice nor three times; the regular octagon's is 135°, and two make 270° while three make 405°. Regular octagons do tile beautifully once squares are allowed in as well, since 135° + 135° + 90° comes to 360° — a mixture known as a semi-regular tessellation.

Irregular shapes that tile anyway

Being regular is not the requirement. Every triangle tessellates, whatever its shape: two copies fit together into a parallelogram, and parallelograms fill the plane in rows.

More surprisingly, every quadrilateral tessellates too, including the ones with a dent in them. Rotating copies half a turn about the midpoints of the sides brings all four different angles round each meeting point, and since the angles of a quadrilateral add to 360°, they close up exactly. Circles, having no straight edges to match, always leave gaps.

Nothing requires the copies to sit the same way up, either. Plenty of tilings depend on turning or flipping alternate tiles, and a shape that refuses to fit in one orientation will often fit in two.

Regular polygons that tessellate on their own: the equilateral triangle, the square and the regular hexagon: 3Regular polygons that tessellate on their own: the equilateral triangle, the square and the regular hexagon

Common questions

Why don't regular pentagons tessellate?

Their interior angle is 108°, and 108 does not divide into 360 a whole number of times. Three pentagons round a point come to 324°, leaving a 36° gap, and a fourth would overlap. Certain irregular pentagons do tile the plane, but no regular one can.

Does every quadrilateral tessellate?

Yes, every single one, convex or concave. Take four copies and rotate each half a turn about the midpoint of a side; the arrangement places all four different angles round each meeting point. Because those four angles add to 360° in any quadrilateral, the tiles close up with no gap.

What is a semi-regular tessellation?

A tiling using more than one kind of regular polygon, with the same arrangement of shapes at every meeting point. The best-known uses regular octagons and squares — 135° + 135° + 90° = 360° — and it is the pattern found on a great many bathroom floors.

Can shapes with curved edges tessellate?

Circles cannot; they always leave gaps between them. But a tile with curved edges can, if the curve cut out of one side is added to the opposite side, so that each bump fits a matching hollow. That is the method behind Escher's interlocking birds and fish.

Last updated August 21, 2026