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# 12.6: Extension: Tessellating Polygons

Difficulty Level: At Grade Created by: CK-12

## 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.

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

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.

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\begin{align*}360^\circ\end{align*}. 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\begin{align*}60^\circ\end{align*}. 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\begin{align*}540^\circ\end{align*} in a pentagon. Each angle in a regular pentagon 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\begin{align*}108^\circ\end{align*} times 2 or 3 does not equal 360\begin{align*}360^\circ\end{align*}.

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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