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Tessellation - Mosinee School District

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9-4 Objectives

You will learn to:

Identify regular tessellations.

Vocabulary

**Tessellation**

Regular **Tessellation**

Uniform

Semi-Regular **Tessellation**

**Tessellation**

A tessellation, or tiling, is a repeating pattern

that completely covers a plane with no gaps or

overlaps. The measures of the angles that meet at

each vertex must add up to 360°.

In the tessellation shown, each

angle of the quadrilateral occurs

once at each vertex. Because

the angle measures of any

quadrilateral add to 360°, any

quadrilateral can be used to tessellate the plane.

Four copies of the quadrilateral meet at each

vertex.

Triangle **Tessellation**

The angle measures of any triangle add up to

180°. This means that any triangle can be used to

tessellate a plane. Six copies of the triangle meet

at each vertex as shown.

Regular

A regular tessellation is formed by congruent

regular polygons.

Regular

tessellation

Regular Polygons

Which regular polygons tessellate

Regular

Polygon

Triangle Square

Pentagon

Hexagon

Heptagon

Octagon

Measure

of one

Interior

Angle

Does it

tessellate

60 90 108 120 128.57 135

yes yes no yes no no

Determine whether a regular 16-gon tessellates the

plane. Explain.

Let

1 represent one interior angle of a regular 16-gon.

m 1

Interior Angle Theorem

Substitution

Simplify.

Answer: Since 157.5 is not a factor of 360,

a 16-gon will not tessellate the plane.

Determine whether a regular 20-gon tessellates the

plane. Explain.

Answer: No; 162 is not a factor of 360.

Uniform

A tessellation pattern can contain any type of polygon.

**Tessellation**s containing the same arrangement of

shapes and angles at each vertex are called uniform.

This tessellation is not uniform. See the different

arrangement of shapes at the different vertexes.

Semiregular

A semiregular tessellation is formed by two

or more different regular polygons, with the

same number of each polygon occurring in the

same order at every vertex.

Semiregular

tessellation

Every vertex has two

squares and three

triangles in this order:

square, triangle,

square, triangle,

triangle.

Determine whether a semi-regular tessellation can be

created from regular nonagons and squares, all having

sides 1 unit long.

Solve algebraically.

Each interior angle of a regular nonagon measures

or 140°.

Each angle of a square measures 90°. Find whole-number

values for n and s such that

All whole numbers greater than 3 will result in a negative

value for s.

Substitution

Simplify.

Subtract from

each side.

Divide each side

by 90.

Answer: There are no whole number values

for n and s so that 140n + 90s = 360.

Determine whether a semi-regular tessellation can be

created from regular hexagon and squares, all having

sides 1 unit long. Explain.

Answer: No; there are no whole number values for h and s

such that 120h + 90s = 360.

STAINED GLASS Stained glass is a very

popular design selection for church and cathedral

windows. It is also fashionable to use stained glass for

lampshades, decorative clocks, and residential

windows. Determine whether the pattern is a

tessellation. If so, describe it as uniform, regular, semiregular,

or not uniform.

Answer:

The pattern is a tessellation

because at the different

vertices the sum of the

angles is 360°. The

tessellation is not uniform

because each vertex does not

have the same arrangement of shapes and angles.

STAINED GLASS Stained glass is a very popular design

selection for church and cathedral windows. It is also

fashionable to use stained glass for lampshades,

decorative clocks, and residential windows. Determine

whether the pattern is a tessellation. If so, describe it

as uniform, regular, semi-regular, or not uniform.

Answer: tessellation, not uniform

What did you learn today

How to:

Identify regular tessellations.

Assignment:

Page 486

11 – 37 odd, 42, 48, 52