Why pentagons can't tile

2026-05-06

Triangles tile. Squares tile. Hexagons tile. Pentagons don't, and the reason isn't subtle - they leave a small triangular gap every time you try.

Drag n. Watch what happens at the central vertex.

At n = 5, three pentagons meet at the centre and stop short of going round. The bottom is left as a thin 36° sliver. Try a fourth pentagon and it overlaps the first - there is no room for it.

Why 360° is the rule

A tiling has to fit perfectly around every vertex. The angles at any single vertex add up to a full turn - 360° exactly, no more and no less. Less, and you get a gap. More, and the shapes overlap.

The interior angle of a regular n-gon is fixed:

angle = (n − 2) · 180° ⁄ n

So for any number of identical regular polygons to fit around a vertex, that angle has to divide 360° a whole number of times.

The three that work

• n = 3 · triangles · interior 60° · six fit, 6 × 60 = 360 ✓
• n = 4 · squares · interior 90° · four fit, 4 × 90 = 360 ✓
• n = 5 · pentagons · interior 108° · 3 × 108 = 324, gap of 36°
• n = 6 · hexagons · interior 120° · three fit, 3 × 120 = 360 ✓
• n = 7 · heptagons · interior ≈ 128.57° · only two fit, gap of 102.9°

Three winners - 3, 4, 6. Pentagons and heptagons fail for opposite reasons: pentagons are almost right (so close that three of them nearly close the loop), heptagons are too wide (only two fit before the third would overlap).

Why the list ends at hexagons

As n grows, the interior angle grows too, creeping towards 180° (a polygon with infinite sides is a straight line). Once the interior angle goes above 120° - which is the moment n goes past 6 - you cannot even fit three copies at a vertex. Three copies need 3 × 120° = 360°; anything more, and they overlap.

So the only candidates that fit three or more copies are n = 3, 4, 5, 6. Of those, pentagon is the only one whose angle isn't a clean divisor of 360. Hence the family ends: 3, 4, 6.

A loophole worth knowing

Regular pentagons can't tile - but irregular ones can. Stretch one corner, squash another, and you can find pentagons that pave a kitchen floor. There are fifteen known families of tiling pentagons; the last one was discovered in 2015. The constraint is the regularity, not the five-sidedness.

And nothing here said the polygons all have to be the same. Mix triangles and hexagons and you get all kinds of beautiful tilings - the Archimedean ones - whose vertex sums also land on 360°, just with a varied diet.

Three winners. One famously close call. The 36° gap that put pentagons out of the running.