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Geometry and Measurement - Tessellations

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A tessellation is a pattern of shapes that fits together perfectly without any gaps or overlaps, covering a flat surface indefinitely. This is also known as 'tiling'. For a tessellation to be possible, the sum of the interior angles of the shapes meeting at any vertex must be exactly 360∘360^\circ.

A simple square tessellation showing four squares meeting at a vertex totaling 360 degrees.
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Regular tessellations are made using only one type of regular polygon. There are only three regular polygons that can form a regular tessellation: the equilateral triangle (6 shapes per vertex), the square (4 shapes per vertex), and the regular hexagon (3 shapes per vertex).

Three regular hexagons meeting at a point, each with 120-degree interior angles.
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Semi-regular (or Archimedean) tessellations are made using two or more different regular polygons. The arrangement of polygons must be identical at every vertex point. For example, a common semi-regular tessellation uses regular octagons and squares.

A semi-regular tessellation showing two regular octagons and one square meeting at a vertex.
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The Interior Angle Rule determines if a regular nn-gon can tessellate. The interior angle must be a divisor of 360∘360^\circ. If the interior angle of a regular polygon is not a factor of 360360, such as the regular pentagon (108∘108^\circ), it cannot form a regular tessellation because 360/108≈3.33360 / 108 \approx 3.33, leaving a gap.

Illustration showing why pentagons overlap or leave gaps when tiled.
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Monohedral tessellations use only one shape, but the shape does not have to be a regular polygon. For example, any triangle or any quadrilateral (even irregular ones) can tile the plane because their internal angles can always be arranged to sum to 360∘360^\circ at a vertex.

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Vertex Notation (or Schläfli symbol variant) is used to describe tessellations. It lists the number of sides of the polygons that meet at a vertex, in order. For a regular hexagon tessellation, the notation is 6.6.66.6.6. For a semi-regular one with two octagons and a square, it is 4.8.84.8.8.

📐Formulae

Interior Angle of a regular n−gon=(n−2)×180∘nInterior\ Angle\ of\ a\ regular\ n-gon = \frac{{(n - 2) \times 180^\circ}}{{n}}

Sum of angles at a vertex=360∘Sum\ of\ angles\ at\ a\ vertex = 360^\circ

Number of shapes at a vertex=360∘Interior AngleNumber\ of\ shapes\ at\ a\ vertex = \frac{{360^\circ}}{{\text{Interior Angle}}}

💡Examples

Problem 1:

Explain why a regular pentagon cannot form a regular tessellation.

Solution:

  1. Calculate the interior angle of a regular pentagon (n=5n = 5): Interior Angle=(5−2)×180∘5=3×180∘5=108∘Interior\ Angle = \frac{{(5 - 2) \times 180^\circ}}{5} = \frac{{3 \times 180^\circ}}{5} = 108^\circ
  2. Check if 108∘108^\circ is a factor of 360∘360^\circ: 360∘108∘≈3.33\frac{{360^\circ}}{{108^\circ}} \approx 3.33

Explanation:

Since 360∘360^\circ divided by 108∘108^\circ does not result in a whole number, the angles cannot meet at a vertex without leaving a gap or overlapping. Therefore, a regular pentagon cannot tessellate.

Problem 2:

Verify if a semi-regular tessellation can be formed by two regular octagons and one square meeting at a vertex.

Solution:

  1. Interior angle of a regular octagon (n=8n=8): Angleoct=(8−2)×180∘8=135∘Angle_{oct} = \frac{{(8 - 2) \times 180^\circ}}{8} = 135^\circ
  2. Interior angle of a square (n=4n=4): Anglesq=90∘Angle_{sq} = 90^\circ
  3. Sum the angles at the vertex: Sum=(2×135∘)+90∘Sum = (2 \times 135^\circ) + 90^\circ Sum=270∘+90∘=360∘Sum = 270^\circ + 90^\circ = 360^\circ

Explanation:

Because the sum of the interior angles of the two octagons and the square is exactly 360∘360^\circ, they can meet at a vertex without gaps or overlaps, forming a semi-regular tessellation.

Problem 3:

Calculate the interior angle of a regular hexagon and determine how many hexagons meet at a single vertex in a regular tessellation.

Solution:

  1. Interior angle for n=6n=6: Angle=(6−2)×180∘6=4×180∘6=120∘Angle = \frac{{(6 - 2) \times 180^\circ}}{6} = \frac{{4 \times 180^\circ}}{6} = 120^\circ
  2. Number of hexagons at a vertex: nhexagons=360∘120∘=3n_{hexagons} = \frac{{360^\circ}}{{120^\circ}} = 3

Explanation:

The interior angle of a regular hexagon is 120∘120^\circ. Since 360/120=3360 / 120 = 3, exactly three hexagons meet at each vertex to form a regular tessellation.

Problem 4:

Determine if a regular dodecagon (12-sided polygon) can form a regular tessellation by itself.

Diagram showing two dodecagons meeting at a vertex with a 60 degree gap remaining.

Solution:

  1. Calculate the interior angle of a regular dodecagon (n=12n=12): Angle=(12−2)×180∘12=1800∘12=150∘\text{Angle} = \frac{{(12 - 2) \times 180^\circ}}{{12}} = \frac{{1800^\circ}}{{12}} = 150^\circ
  2. Check if 360∘360^\circ is divisible by 150∘150^\circ: 360∘150∘=2.4\frac{{360^\circ}}{{150^\circ}} = 2.4
  3. Since 2.4 is not an integer, the angles cannot sum to 360∘360^\circ at a vertex without leaving a gap or overlapping.

Explanation:

For a regular polygon to tessellate alone, its interior angle must be a factor of 360∘360^\circ. A dodecagon's angle of 150∘150^\circ results in a gap of 360−(2×150)=60∘360 - (2 \times 150) = 60^\circ, which is not enough space for a third dodecagon.

Problem 5:

A semi-regular tessellation is made using regular hexagons and equilateral triangles. If two hexagons meet at a vertex, how many equilateral triangles are required to complete the vertex?

Vertex diagram showing two 120 degree angles and two 60 degree angles summing to 360.

Solution:

  1. Find the interior angle of a regular hexagon: Hexagon Angle=(6−2)×180∘6=120∘\text{Hexagon Angle} = \frac{{(6 - 2) \times 180^\circ}}{{6}} = 120^\circ
  2. Calculate the sum of the angles for two hexagons: 120∘+120∘=240∘120^\circ + 120^\circ = 240^\circ
  3. Calculate the remaining angle needed to reach 360∘360^\circ: 360∘−240∘=120∘360^\circ - 240^\circ = 120^\circ
  4. Find the interior angle of an equilateral triangle: Triangle Angle=(3−2)×180∘3=60∘\text{Triangle Angle} = \frac{{(3 - 2) \times 180^\circ}}{{3}} = 60^\circ
  5. Divide the remaining angle by the triangle's angle: 120∘60∘=2\frac{{120^\circ}}{{60^\circ}} = 2 Therefore, 2 equilateral triangles are needed.

Explanation:

To fill the 360∘360^\circ around a vertex, the combined angles of the hexagons (240∘240^\circ) must be supplemented by 120∘120^\circ worth of triangles. Since each triangle provides 60∘60^\circ, two triangles are required.