5-9 Tessellations
Learn to predict and verify patterns
involving tessellations.
Pre-Algebra
Vocabulary
tessellation
regular tessellation
semiregular tessellation
In a regular tessellation, a regular
polygon is repeated to fill a plane. The
angles at each vertex add to 360°, so
exactly three regular tessellations exist.
Fascinating designs can be made by
repeating a figure or group of figures.
These designs are often used in art and
architecture.
A repeating pattern of plane figures
that completely covers a plane with no
gaps or overlaps is a tessellation.
In a semiregular tessellation, two or
more regular polygons are repeated to fill
the plane and the vertices are all identical.
1
Additional Example 1: Problem Solving Application
Find all the possible semiregular
tessellations that use triangles and
squares.
1
Understand the Problem
List the important information:
• The angles at each vertex add to 360°.
• All the angles in a square measure 90°.
• All the angles in an equilateral triangle
measure 60°.
Additional Example 1 Continued
2
Account for all possibilities: List all possible
combinations of triangles and squares
around a vertex that add to 360°. Then see
which combinations can be used to create
a semiregular tessellation.
6 triangles, 0 squares 6(60°) = 360° regular
3 triangles, 2 squares 3(60°) + 2(90°) = 360°
0 triangles, 4 squares 4(90°) = 360° regular
Additional Example 1 Continued
3
Solve
There are two arrangements of three
triangles and two squares around a
vertex.
Additional Example 1 Continued
3
Solve
Repeat each arrangement around
every vertex, if possible, to create a
tessellation.
Additional Example 1 Continued
3
Solve
There are exactly
two semiregular
tessellations that
use triangles and
squares.
Make a Plan
Additional Example 1 Continued
4
Look Back
Every vertex in each arrangement
is identical to the other vertices in
that arrangement, so these are the
only arrangements that produce
semiregular tessellations.
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Additional Example 2: Creating a Tessellation
Try This: Example 2
Create a tessellation with quadrilateral
EFGH.
Create a tessellation with quadrilateral IJKL.
J
There must be a
copy of each angle
of quadrilateral
EFGH at every
vertex.
K
L
I
There must be a copy of each angle of
quadrilateral IJKL at every vertex.
Additional Example 3: Creating a Tessellation by
Transforming a polygon
Use rotations to create a tessellation
with the quadrilateral given below.
Additional Example 3 Continued
Step 5: Use the figure to make a tessellation.
Step 1: Find the midpoint of a side.
Step 2: Make a new edge for half of the side.
Step 3: Rotate the new edge around the
midpoint to form the edge of the other half
of the side.
Step 4: Repeat with the other sides.
Try This: Example 3
Use rotations to create a tessellation
with the quadrilateral given below.
Try This: Example 3 Continued
Step 5: Use the figure to make a tessellation.
Step 1: Find the midpoint of a side.
Step 2: Make a new edge for half of the side.
Step 3: Rotate the new edge around the
midpoint to form the edge of the other half
of the side.
Step 4: Repeat with the other sides.
3