Regular Polygon Tessellations

You will need to use some of the pattern blocks located in our classroom to build some of the examples on this page.

Make patterns with some pattern blocks, making certain that you leave no gaps or spaces.

Which shapes fit together easily?
Which shapes don't seem to fit with the others?
Which patterns could be repeated over and over again in the plane?
What shapes fit together making a pattern using only one type of block?
What shapes fit together making a pattern using two blocks that are different?
What shapes fit together making a pattern using three blocks that are different?

You should also visit the Tessellation Tool at:
which will allow you to build tessellations and other designs by attaching the corners of various shapes.

A regular polygon has 3 or 4 or 5 or more sides and angles, all equal.
 A regular tessellation means a tessellation made up of congruent regular polygons.

Note: Regular means that the sides of the polygon are all the same length.
 Congruent means that the polygons are all the same size and shape.

A polygon is a many-sided shape. A regular polygon is one in which all of the sides and angles are equal. Some examples are shown below.

Name the regular polygons shown above.

A vertex is a point at which three or more tiles in a tessellation meet. Two tiles cannot meet in a point, but have to meet in line.

Using pattern blocks, try tessellating with hexagons as shown below.

How many hexagons meet at each vertex?

Next, try tessellating with squares.

How many square tiles meet at each vertex?

Finally, using the pattern blocks, try tessellating with triangles. 

How many regular triangles meet at each vertex?

Here is an example of pentagons.

Are the pentagons tessellating?  Explain your reasoning.

a tessellation of triangles

a tessellation of squares
a tessellation of hexagons

When you look at the three examples above, you can easily see that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide is involved.

Complete the following chart.

Regular Polygon
Measure of each
 interior angle
360 degrees ÷
by the measurement
of the interior angle
Will the polygon
tessellate using only
this polygon?
equilateral triangle 60 360 ÷ 60 = 6
square 90
regular pentagon  
regular hexagon  
regular heptagon  
regular octagon  

Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.

In a tessellation the polygons used will fit together with their angles arranged around a point with no gaps or overlaps. When using just one polygon (for example, only equilateral triangles), the interior measure of each angle will need to be a factor of _____ degrees (meaning that ____ degrees can be divided evenly by that angle measure). The only regular polygons that qualify are the __________________, ___________________, and ___________________.

Choose any polygon other than a triangle, square, or hexagon. Illustrate and explain why it will not tessellate regularly in the Euclidean plane.

What is a regular tessellation?

Visit Hyperbolic Tessellations to help you find out:
Be sure to try out the Java Applet to create your own hyperbolic tessellation.

Our Tessellation WebQuest

Alice Drive Middle School

Our Tessellation WebQuest developed by Cynthia R. Parker
7th Grade Mathematics Instructor
Alice Drive Middle School
Sumter School District #17
Sumter, South Carolina
July 2004