Tessellation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Tessellation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A tessellation is a pattern that covers an infinite plane with repeated geometric shapes, leaving no gaps and having no overlaps.

Like a bathroom floor tile pattern that fits together perfectly and could extend forever in all directions.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A regular polygon tessellates the plane only if its interior angle evenly divides 360°; only triangle, square, and hexagon work.

Common stuck point: Students overlook tiny gaps or overlaps in repeated patterns.

Sense of Study hint: Check angle sums around each vertex to confirm full 360^circ coverage.

Worked Examples

Example 1

medium
Explain why regular hexagons tessellate the plane but regular pentagons do not.

Solution

  1. 1
    For a regular polygon to tessellate alone, its interior angle must divide 360° evenly.
  2. 2
    Interior angle of a regular hexagon: \dfrac{(6-2)\times180°}{6} = 120°. And 360° \div 120° = 3 (whole number). So exactly 3 hexagons meet at each vertex. ✓
  3. 3
    Interior angle of a regular pentagon: \dfrac{(5-2)\times180°}{5} = 108°. And 360° \div 108° = 3.\overline{3} (not a whole number). ✗
  4. 4
    Since 3.\overline{3} pentagons cannot fit exactly around a vertex, regular pentagons do not tessellate.

Answer

Regular hexagons tessellate (interior angle 120° divides 360° exactly); regular pentagons do not (interior angle 108° does not).
The vertex condition for a regular polygon tessellation requires that the interior angle divides 360° evenly. Only equilateral triangles (60°), squares (90°), and regular hexagons (120°) satisfy this among all regular polygons.

Example 2

hard
A proposed semi-regular tiling places 2 triangles and 2 squares at every vertex. Verify whether the vertex angles sum to 360°, and if not, find an arrangement of triangles and squares that does work.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Does an equilateral triangle tessellate the plane? Justify your answer using the interior angle.

Example 2

medium
A tiling uses regular octagons and squares. One proposed vertex arrangement has 1 octagon and 2 squares. Does this satisfy the 360° vertex condition? What arrangement actually works?
← Back to Tessellation Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

tiling intuitionpolygon generalsymmetry