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12.6: Extension: Tessellating Polygons

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Learning Objectives

  • Tessellating regular polygons

What is a Tessellation?

You have probably seen tessellations before. Examples of a tessellation are: a tile floor, a brick or block wall, a checker or chess board, and a fabric pattern.



We are only going to worry about tessellating regular polygons. To tessellate a shape it must be able to exactly surround a point, or the sum of the angles around each point in a tessellation must be 360^\circ. The only regular polygons with this feature are equilateral triangles, squares, and regular hexagons.

Example 1: Draw a tessellation of equilateral triangles.

Solution: In an equilateral triangle each angle is 60^\circ. Therefore, six triangles will perfectly fit around each point.

Extending the pattern, we have:

Example 2: Does a regular pentagon tessellate?

Solution: First, recall that there are 540^\circ in a pentagon. Each angle in a regular pentagon is 540^\circ \div 5 = 108^\circ. From this, we know that a regular pentagon will not tessellate by itself because 108^\circ times 2 or 3 does not equal 360^\circ.

Tessellations can also be much more complicated. Check out http://www.mathsisfun.com/geometry/tessellation.html to see other tessellations and play with the Tessellation Artist, which has a link at the bottom of the page.

Review Questions

  1. You were told that equilateral triangles, squares, and regular hexagons are the only regular polygons that tessellate. Tessellate a square. Add color to your design.
  2. What is an example of a tessellated square in real life?
  3. How many regular hexagons will fit around one point? (First, recall how many degrees are in a hexagon, and then figure out how many degrees are in each angle of a regular polygon. Then, use this number to see how many of them fit around a point.)
  4. Using the information from #2, tessellate a regular hexagon. Add color to your design.
  5. You can also tessellate two regular polygons together. Try tessellating a regular hexagon and an equilateral triangle. First, determine how many of each fit around a point and then repeat the pattern. Add color to your design.

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Date Created:

Feb 22, 2012

Last Modified:

Aug 21, 2014
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