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.

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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 tessellation repeats one or more shapes across the whole plane so that around every meeting point the angles add to exactly 360°360°.

Common stuck point: The procedure for tessellation is the easy part; the trap is thinking every regular polygon tessellates. Asking "Do copies of the shape fill the flat surface completely with no gaps and no overlaps?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do copies of the shape fill the flat surface completely with no gaps and no overlaps?

Worked Examples

Example 1

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

Answer

Regular hexagons tessellate (interior angle 120°120° divides 360°360° exactly); regular pentagons do not (interior angle 108°108° does not).

First step

1
For a regular polygon to tessellate alone, its interior angle must divide 360°360° evenly.

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Example 2

hard
A proposed semi-regular tiling places 22 triangles and 22 squares at every vertex. Verify whether the vertex angles sum to 360°360°, and if not, find an arrangement of triangles and squares that does work.

Example 3

medium
Check the vertex configuration 3.6.3.63.6.3.6: equilateral triangle, hexagon, triangle, hexagon. Does it sum to 360°360°?

Example 4

medium
A vertex meets a square (90°90°), a regular hexagon (120°120°), and a third regular polygon. What is the interior angle of the third polygon?

Example 5

medium
Solve for nn: a regular nn-gon tiles the plane alone if and only if 2nn−2\tfrac{2n}{n-2} is a positive integer. List the valid nn.

Example 6

hard
A vertex configuration 4.6.124.6.12 proposes a square, a hexagon, and a regular dodecagon meeting at one vertex. Verify it sums to 360°360°.

Example 7

hard
Count distinct semiregular (Archimedean) tilings of the plane.

Example 8

challenge
Penrose tilings use two prototiles (kite and dart) to cover the plane aperiodically. Why is this impossible with a single regular polygon?

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 11 octagon and 22 squares. Does this satisfy the 360°360° vertex condition? What arrangement actually works?

Example 3

easy
What must the interior angles at each vertex of a tessellation sum to?

Example 4

easy
What is the interior angle of a regular pentagon?

Example 5

easy
A tile leaves visible gaps between copies. Is the pattern a tessellation?

Example 6

easy
A tile in a tessellation overlaps part of its neighbor. Is this allowed?

Example 7

medium
Check whether two regular hexagons and one equilateral triangle can meet at a single vertex.

Example 8

medium
Are all quadrilaterals (including non-convex ones) able to tile the plane?

Example 9

medium
What is the interior angle of a regular 99-gon, and does it tile the plane alone?

Example 10

medium
In a square tiling, what transformation sends one square to a neighboring square?

Example 11

hard
A vertex configuration 3.7.423.7.42 sums to 360°360° algebraically. Why is it nonetheless not a true vertex of any tiling?

Example 12

hard
A floor is to be tiled with regular polygons so the same configuration appears at every vertex. Why must only one of 3.3.3.3.63.3.3.3.6 and 6.3.3.3.36.3.3.3.3 list correctly?

Example 13

hard
In the hexagonal tiling, three hexagons meet at every vertex. How many vertices does each hexagon contribute, and how many vertices belong to each hexagon counted per-hexagon?

Example 14

challenge
A soccer ball is built from regular pentagons (1212) and regular hexagons (2020). Using Euler's formula V−E+F=2V-E+F=2, verify the count of pentagons.
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Background Knowledge

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

tiling intuitionpolygon generalsymmetry