Tessellations in geometry

A couple of examples of tessellations in geometry are shown below. Basically, whenever you place a polygon together repeatedly without any gaps or overlaps, the resulting figure is a tessellation. A tessellation is also called tiling. Of course, tiles in your house is a real-life example of tessellation.

Tessellation with rectangles

Tessellation made with rectangles

Tessellation with equilateral triangles

Tessellation made with triangles

Tessellations in geometry: How to quickly check if a geometric figure will tessellate.


What makes the above figures tessellations?

1) The rectangles or triangles are repeated to cover a flat surface.

2) No gaps, or overlaps between the rectangles or the triangles.

Not all figures will form tessellations in geometry.


When a figure can form a tessellation, the figure is said to tessellate.


Every triangle tessellates


Every quadrilateral tessellates


Why can we say with confidence that the above 2 statements are true?

Well, since there are no gaps and no overlaps, the sum of the measures of the angles around any vertex must be equal to 360 degrees as seen below with red circles.


360 degrees angle in a  tessellation
360 degrees angle in a tesselation

Therefore, if the measure of an angle of a figure is not a factor of 360, it will not tessellate. This is a very important concept since it will help determine when a figure can tessellate.

We can use then the formula to find the interior angle of a regular polygon to check if a figure will tessellate.

Interior angle of a regular polygon = [180 × (n-2)] / n

A couple of examples showing how to use the formula above to determine which figures will tessellate.


Example #1

Determine whether a regular pentagon will tessellate

A pentagon has 5 sides, so n = 5.

Interior angle of the pentagon = [180 × (5-2)] / 5

Interior angle of the pentagon = [180 × 3] / 5

Interior angle of the pentagon = 540 / 5

Interior angle of the pentagon = 108 degrees

There is no way to make 360 with 108 since 108 + 108 + 108 = 324 and 108 + 108 + 108 + 108 = 432

Therefore the pentagon will not tessellate as you can see below:

Example of an image that is not a tessellation


The gap is shown with a red arrow!

Example #2

Determine whether a regular 16-gon will tessellate

An 16-gon has 16 sides, so n = 16.

Interior angle of the 16-gon = [180 × (16-2)] / 16

Interior angle of the 16-gon = [180 × 14] / 16

Interior angle of the 16-gon = 2520 / 16

Interior angle of the 16-gon = 157.5 degrees

157.5 is not a factor of 360, so the 16-gon will not tessellate.

Tessellations can happen with translations, rotations, and reflections and it can also happen with irregular figures.

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