Which regular polygons can tile the plane — and why the interior angle is the deciding factor.
The Mnemonic
Covering a Plane With No Gaps or Overlaps
A tessellation covers an entire flat plane using repeated copies of a shape, with absolutely no gaps and no overlaps. Tessellations are built using translations, rotations, and reflections (the isometries) to repeat a shape across the whole plane.
For a REGULAR polygon to tessellate by itself, the interior angle must divide evenly into 360° — since several copies of that same angle must fit perfectly around every single vertex where the shapes meet. Only THREE regular polygons satisfy this: the equilateral triangle (60°, and 6 fit around a point: 6×60°=360°), the square (90°, and 4 fit: 4×90°=360°), and the regular hexagon (120°, and 3 fit: 3×120°=360°).
A tessellation covers a plane with no gaps and no overlaps — the angles meeting at each vertex must add to exactly 360°
💡 Memory Trick
"Which regular polygons can tile the plane — and why." Only three: TRIANGLE, SQUARE, HEXAGON. The test is simple: does 360° divide evenly by that polygon's interior angle? Triangle (60°): 360/60=6 ✓. Square (90°): 360/90=4 ✓. Pentagon (108°): 360/108=3.33... ✗ (doesn't divide evenly — this is exactly why pentagons can't tile a floor by themselves).
Why It Works
Connecting Directly to Polygon Angle Sum and Regular Polygons
This entire topic is really just an application of the interior angle formula from Regular Polygons: each interior angle equals [(n−2)×180°]/n. For a shape to tessellate by itself, whole copies of that interior angle must exactly fill the 360° available around each vertex point — meaning 360° divided by the interior angle must come out to a whole number, with no leftover angle and no overlap.
This is exactly why only three regular polygons work: as the number of sides increases past 6, the interior angle keeps growing (getting closer to 180°), and 360° stops dividing evenly by it — there's a genuine mathematical cutoff, not just a coincidence of which shapes "happen" to look like they'd fit.
Using It In A Proof
Testing Whether a Regular Polygon Tessellates
Testing any regular polygon (not just the three that work) follows the same mechanical check every time.
1
Find the polygon's interior angle
Use the Regular Polygons formula: interior angle = [(n−2)×180°]/n.
2
Divide 360° by that interior angle
Check whether 360° ÷ (interior angle) gives a whole number with no remainder.
3
Conclude
Whole number result → the polygon tessellates by itself. Non-whole-number result → it does not.
Full Worked Example
Testing a Regular Octagon for Tessellation
Given: A regular octagon (8 sides). Determine: whether it can tessellate a plane by itself.
Since 2.67 is not a whole number, a regular octagon CANNOT tessellate a plane by itself.
This confirms the familiar real-world observation: octagon floor tiles are always combined with small squares filling the gaps — a pure octagon tessellation with no other shape is mathematically impossible.
🎯 Quick Worked Example
Confirm whether a regular hexagon can tessellate by checking its interior angle divides evenly into 360°.
Conclude. Since 3 is a whole number, the regular hexagon DOES tessellate — exactly 3 hexagons meet perfectly at every vertex, matching the honeycomb pattern seen in nature.
📌 Exam Application
Beyond just the three regular polygons that tessellate alone, some exam questions ask about COMBINATIONS of different regular polygons meeting at one vertex (like octagons plus squares) — the same core rule still applies: whatever combination of angles meets at a single vertex must sum to exactly 360°, whether from one polygon type or several mixed together.
⚠️ Most Common Tessellations Mistakes
Trap 1 — Assuming any regular polygon can tessellate: Only three regular polygons (triangle, square, hexagon) can tile a plane alone — the vast majority of regular polygons (pentagons, heptagons, octagons, etc.) cannot, since their interior angles don't divide evenly into 360°.
Trap 2 — Forgetting to check for a whole-number result specifically: A 'close' result like 3.33 or 2.67 is NOT good enough — the division must come out to an exact whole number, since a partial angle would leave a gap or force an overlap at that vertex.
✓ Quick Self-Test
1) What condition must be true for a regular polygon to tessellate by itself? 2) Which three regular polygons satisfy this condition? 3) Why can't a regular pentagon tessellate by itself? 4) Verify that a regular equilateral triangle satisfies the tessellation condition. 5) How does the tessellation rule connect to the interior angle formula from Regular Polygons?