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# 12.6: Extension: Tessellations

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Determine whether or not a given shape will tessellate.

## What is a Tessellation?

You have probably seen tessellations before, even though you did not call them that. Examples of a tessellation are: a tile floor, a brick or block wall, a checker or chess board, and a fabric pattern.

Tessellation: A tiling over a plane with one or more figures such that the figures fill the plane with no overlaps and no gaps.

Here are a few examples.

Notice the hexagon (cubes, first tessellation) and the quadrilaterals fit together perfectly. If we keep adding more, they will entirely cover the plane with no gaps or overlaps. The tessellation pattern could be colored creatively to make interesting and/or attractive patterns.

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\begin{align*}360^\circ\end{align*}. Therefore, every quadrilateral and hexagon will tessellate.

Example 1: Tessellate the quadrilateral below.

Solution: To tessellate any image you will need to reflect and rotate the image so that the sides all fit together. First, start by matching up each side with itself around the quadrilateral.

This is the final tessellation. You can continue to tessellate this shape forever.

Now, continue to fill in around the figures with either the original or the rotation.

Example 2: Does a regular pentagon tessellate?

Solution: First, recall that there are (52)180=540\begin{align*}(5 - 2)180^\circ = 540^\circ\end{align*} in a pentagon and each angle is 540÷5=108\begin{align*}540^\circ \div 5 = 108^\circ\end{align*}. From this, we know that a regular pentagon will not tessellate by itself because 108×3=324\begin{align*}108^\circ \times 3 = 324^\circ\end{align*} and 108×4=432\begin{align*}108^\circ \times 4 = 432^\circ\end{align*}.

For a shape to be tessellated, the angles around every point must add up to 360\begin{align*}360^\circ\end{align*}. A regular pentagon does not tessellate by itself. But, if we add in another shape, a rhombus, for example, then the two shapes together will tessellate.

Tessellations can also be much more complicated. Here are a couple of examples.

## Review Questions

Will the given shapes tessellate? If so, make a small drawing on grid paper to show the tessellation.

1. A square
2. A rectangle
3. A rhombus
4. A parallelogram
5. A trapezoid
6. A kite
8. Which regular polygons will tessellate?
9. Use equilateral triangles and regular hexagons to draw a tessellation.
10. The blue shapes are regular octagons. Determine what type of polygon the white shapes are. Be as specific as you can.
11. Draw a tessellation using regular hexagons.
12. Draw a tessellation using octagons and squares.
13. Make a tessellation of an irregular quadrilateral using the directions from Example 1.

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