POLYGONS

and

TESSELLATIONS

By

Citra Triwana Marpaung

ACA 111 0050

Angles in Polygons

The vertex angles of a polygons with four or more sides can be any size between 0˚ and 360˚.

180˚

360˚

Example

In the hexagon figure, we

can known that,

B is less than 20˚ and

D and A are both

greather than 180˚.

In spite of this range of possible size, there is a relationship

between the sum of all the angles in polygon and its number sides.

In any triangle, the sum of the three angle

measures is 180˚. One way of demostrating that

theorem is to draw an abitrary triangle and cut off

its angles.

The sum of the angles in a polygon with four or more sides can

be found by subdividing the polygon into triangels so that the

vertices of the triangles are the vertice of the polygon.

1

2

3

4

5

6

In quadrilateral in that figure above is partitioned into two triangles.

Thus, the sum of all angles (six angel of both the triangles) is

2 triangles x 180˚ = 360˚

However, since each the quadrilateral can be partitioned into

triangles such that the vertices of triangles are also the vertices of

a quadrilateral, the sum of the angles a quadrilateral will always

be 360˚.

Similar approach can be used to find the sum of the angles in any

polygon.

The total number

of degrees in its

angles is

3 x 180˚ = 540˚

Congruence

The idea of Congruence is one figure can be placed on the

other, so that they coincide.

Another way to describe congruent plane figures is to say

that they have same size and shape.

Two line segments are congruent if they have the same

length. And two angles are congruent if they have same

measure..

Regular Polygons

The figures in those photographs are examples of regular polygons.

A polygon is called a regular polygon if it satisfies

both of the folloeing conditions:

1. All angles are congruent.

2. All sides are congruent

Regular pentagon Equilateral triangle square

Drawing Regular Polygons

Three special angles in regular polygons.

A vertex angle

is formed by two adjacent sides of polygon

A central angle

is formed by connecting the centerof

the polygon to two adjacent vertices

of polygon

is formed by one side of the polygon

and the extension of an adjacent

sides.

Exterior angle

The sum of the measures of the angles in a polygon can be used

to compute the number of degrees in each vertex angle of a

regular polygon; Simply divide the sum of all the measures of the

angles by the number of angles.

The sum angles in pentagon is

3 triangles x 180˚ = 540˚

Therefore, each angle in a regular

pentagon is

540˚ 5 vertex angles = 108˚

108˚

108˚

108˚108˚

108˚

Steps of Drawing a Regular Polygon

Step(1) Draw a line segment

and mark a vertex

Step(2) Measure off a 108˚ angle

Step(3) Mark off two sides of

equal length.

108˚

108˚

Step(4) Measure off a second

angle of 108˚

Antother approach to drawing regular polygons begins with a

circle and uses central angles.

The number of degrees in the central angle of a regular polygon

is 360˚ devided by the number of sides in the polygon.

Decagon

36˚

A decagonhas 10 sides, then each

central angle is 360˚ 10 = 36˚

36˚

Tessellation With Polygons

Any arrangement in which no overlapping figures are

placed together to entirely cover a region is called a

tessellation.

Floors and ceilings are often tesselated with square-

shaped material, because squares can be joined together

without gaps or overlaps

A regular hexagon

Squares

Equilateral Triangle

These three types that will tessellate by themselves

Tessellation with regular hexagon

The points at which the vertices of the hexagon meet are the vertex points of the tessellation.

12

34

5

A tessellation of two or more noncongruent regular polygons in which

each vertex is surrounded by the same arrangement of polygons, called

SEMIREGULAR

SEMIREGULAR NOT SEMIREGULAR

Problem Solving Application

Consider a regular polygon with fewer sides. A regular hexagon has six congruent vertex angle, and since it can be partitioned into four triangles.

What is the number of degrees in one of its vertex angles?

What is the size of each vertex angles in hexagon?

4 x 180˚ = 720˚

720˚ 6 = 120˚