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In this presentation:

We will be looking at different aspects of geometry like symmetry, analyzing geometric shapes, and more in the different sections that will follow.

We will be looking at different aspects of geometry like symmetry, analyzing geometric shapes, and more in the different sections that will follow.

1.
Math 126

Chapter 12

Geometric Shapes

Chapter 12

Geometric Shapes

2.
Section 12.1 Recognizing Shapes

The van Hiele Theory

Level 0 (Recognition)

At this lowest level, a child recognizes certain shapes holistically (Greek “holos”

means whole) without paying attention to their components.

In other words, a child may recognize a square without knowing the meaning of a

square; in the same way that a child recognizes an avocado without knowing the

definition of an avocado.

Level 1 (Analysis)

The child starts to focus analytically on the parts of a figure, such as its sides and

angles. Component parts and their attributes are used to describe and characterize

figures. Relevant attributes are understood and are differentiated from irrelevant

In other words, a child starts to understand the definition of a square by knowing that

a square must have exactly 4 congruent sides and 4 right angles.

The van Hiele Theory

Level 0 (Recognition)

At this lowest level, a child recognizes certain shapes holistically (Greek “holos”

means whole) without paying attention to their components.

In other words, a child may recognize a square without knowing the meaning of a

square; in the same way that a child recognizes an avocado without knowing the

definition of an avocado.

Level 1 (Analysis)

The child starts to focus analytically on the parts of a figure, such as its sides and

angles. Component parts and their attributes are used to describe and characterize

figures. Relevant attributes are understood and are differentiated from irrelevant

In other words, a child starts to understand the definition of a square by knowing that

a square must have exactly 4 congruent sides and 4 right angles.

3.
Section 12.1 Recognizing Shapes

The van Hiele Theory

Level 2 (Relationships)

There are two types of thinking at this level. First, a child understands abstract

relationships among figures. For example, a square is both a rhombus and a rectangle.

Second, a child can use informal deductions to justify observations made at level 1.

For instance, a rhombus is also a parallelogram; or a square is also a rectangle.

Level 3 (Deduction)

Reasoning at this level includes the study of geometry as a formal mathematical

system. A student at this level can use logical reasonings to make deductions.

For example, to deduce that the base angles of an isosceles triangle must be congruent.

A student at this level can also understand the notions of mathematical postulates

(axioms) and theorems, such as the Pythagorean theorem.

The van Hiele Theory

Level 2 (Relationships)

There are two types of thinking at this level. First, a child understands abstract

relationships among figures. For example, a square is both a rhombus and a rectangle.

Second, a child can use informal deductions to justify observations made at level 1.

For instance, a rhombus is also a parallelogram; or a square is also a rectangle.

Level 3 (Deduction)

Reasoning at this level includes the study of geometry as a formal mathematical

system. A student at this level can use logical reasonings to make deductions.

For example, to deduce that the base angles of an isosceles triangle must be congruent.

A student at this level can also understand the notions of mathematical postulates

(axioms) and theorems, such as the Pythagorean theorem.

4.
Section 12.1 Recognizing Shapes

The van Hiele Theory

Level 4 (Axiomatics)

Geometry at this level is highly abstract and does not necessarily involve concrete or

pictorial models. The postulates or axioms themselves become the object of intense,

rigorous study.

Examples are

a. high dimensional geometry, where we cannot make a concrete model of an n-

dimensional sphere or cube.

b. Non- Euclidean geometry, where lines are not really straight in usual sense, and

non-Euclidean lines are called “geodesics”.

This level of study is only suitable for university or even graduate students.

The van Hiele Theory

Level 4 (Axiomatics)

Geometry at this level is highly abstract and does not necessarily involve concrete or

pictorial models. The postulates or axioms themselves become the object of intense,

rigorous study.

Examples are

a. high dimensional geometry, where we cannot make a concrete model of an n-

dimensional sphere or cube.

b. Non- Euclidean geometry, where lines are not really straight in usual sense, and

non-Euclidean lines are called “geodesics”.

This level of study is only suitable for university or even graduate students.

5.
Level 0 (Recognition)

Children at this level will learn to associate the names “triangle, rectangle,

square” with the appropriate shape, without necessarily understanding the

meaning of the words.

They may just recognize a triangle because it is similar to the one they saw

in the classroom. They may not know that triangle is a shape with 3

This is similar to recognizing their teacher or principal.

Consequently, if the shape is drawn upside down, or drawn in different

colors, or in different sizes, the child may not recognize it as a triangle any

Children at this level will learn to associate the names “triangle, rectangle,

square” with the appropriate shape, without necessarily understanding the

meaning of the words.

They may just recognize a triangle because it is similar to the one they saw

in the classroom. They may not know that triangle is a shape with 3

This is similar to recognizing their teacher or principal.

Consequently, if the shape is drawn upside down, or drawn in different

colors, or in different sizes, the child may not recognize it as a triangle any

6.
To test their comprehension,

children at this level will be asked to separate the above shapes into

3 categories: Triangles, rectangles, and squares.

children at this level will be asked to separate the above shapes into

3 categories: Triangles, rectangles, and squares.

7.
Similarly, they will be asked to sort out the circles or ovals

from the following diagram.

from the following diagram.

8.
Identify the triangles, squares, and rhombi in this quilt.

9.
Section 12.2

Analyzing Geometric Shapes – Level 1

At this level, children will start to understand the meaning

of triangles, different types of quadrilaterals, parallel and

perpendicular lines.

The word “Triangle” will be treated as a compound word

“tri-angle”, where “tri” means 3 in Latin; and “angle”

means corners.

The word “Quadrilateral” will also be explained as the

compound word “quadri-lateral”, where “quadri” means 4

in Latin, and “lateral” means side.

(note: “quadra” means “square” in Latin, hence quadratic equation means degree 2 equation.)

Analyzing Geometric Shapes – Level 1

At this level, children will start to understand the meaning

of triangles, different types of quadrilaterals, parallel and

perpendicular lines.

The word “Triangle” will be treated as a compound word

“tri-angle”, where “tri” means 3 in Latin; and “angle”

means corners.

The word “Quadrilateral” will also be explained as the

compound word “quadri-lateral”, where “quadri” means 4

in Latin, and “lateral” means side.

(note: “quadra” means “square” in Latin, hence quadratic equation means degree 2 equation.)

10.
Symmetries

In formal terms, we say that an object is symmetric with respect to a given

mathematical operation, if, when applied to the object, this operation does

not change the object or its appearance.

Reflection Symmetry (also called folding symmetry)

A 2D figure has reflection symmetry if there is a line that the figure can be

“folded over” so that one-half of the figure matches the other half

The “fold line” just described is call the figure’s line (axis) of symmetry.

In formal terms, we say that an object is symmetric with respect to a given

mathematical operation, if, when applied to the object, this operation does

not change the object or its appearance.

Reflection Symmetry (also called folding symmetry)

A 2D figure has reflection symmetry if there is a line that the figure can be

“folded over” so that one-half of the figure matches the other half

The “fold line” just described is call the figure’s line (axis) of symmetry.

11.
Lines of symmetry for the following common figures.

12.
Rotation Symmetry

A 2D figure has rotation symmetry if there is a point around which the

figure can be rotated, less than a full turn, so that the image matches the

original figure perfectly.

(click to see animation)

This equilateral triangle has 2 (non-trivial) rotation symmetries, 120° and

240° respectively. Since every figure will match itself after rotating 360°,

we do not consider a 360° rotation as a rotation symmetry.

A 2D figure has rotation symmetry if there is a point around which the

figure can be rotated, less than a full turn, so that the image matches the

original figure perfectly.

(click to see animation)

This equilateral triangle has 2 (non-trivial) rotation symmetries, 120° and

240° respectively. Since every figure will match itself after rotating 360°,

we do not consider a 360° rotation as a rotation symmetry.

13.
Rotation symmetries of common figures

We don’t count the trivial 360° rotation symmetry here.

Rectangle Square

(1 symmetry) (3 symmetries) Diamond

(1 symmetry)

(1 symmetry) Trapezoid

Regular Pentagon (no symmetry)

(4 symmetries)

We don’t count the trivial 360° rotation symmetry here.

Rectangle Square

(1 symmetry) (3 symmetries) Diamond

(1 symmetry)

(1 symmetry) Trapezoid

Regular Pentagon (no symmetry)

(4 symmetries)

14.
Informal Definition of common shapes

Model Abstraction Description

Top of a window Line segment

Open pair of scissors Angle The union of two line

segments with a

common endpoint.

Vertical flag pole Right angle Angle formed by two

lines, one vertical

and one horizontal.

Model Abstraction Description

Top of a window Line segment

Open pair of scissors Angle The union of two line

segments with a

common endpoint.

Vertical flag pole Right angle Angle formed by two

lines, one vertical

and one horizontal.

15.
Model Abstraction Description

Railroad tracks Parallel lines Two distinct line

segments which, if

extended in both

directions, never

meet.

Railroad tracks Parallel lines Two distinct line

segments which, if

extended in both

directions, never

meet.

16.
Model Abstraction Description

Bike frame Scalene triangle Triangle with 3 sides

of different lengths

Pennant Isosceles triangle Triangle with at least

two sides of the

same length

Yield sign Equilateral triangle Triangle with three

sides of the same

length

Steel frame of a Right triangle Triangle with one

bridge right angle.

Bike frame Scalene triangle Triangle with 3 sides

of different lengths

Pennant Isosceles triangle Triangle with at least

two sides of the

same length

Yield sign Equilateral triangle Triangle with three

sides of the same

length

Steel frame of a Right triangle Triangle with one

bridge right angle.

17.
Model Abstraction Description

Railing Parallelogram Quadrilateral with

parallel opposite

sides

Car jack Rhombus Quadrilateral with

four sides equal in

length.

Door Rectangle Quadrilateral with

four right angles

Floor tile Square Quadrilateral with

four sides equal in

length and four right

angles.

Railing Parallelogram Quadrilateral with

parallel opposite

sides

Car jack Rhombus Quadrilateral with

four sides equal in

length.

Door Rectangle Quadrilateral with

four right angles

Floor tile Square Quadrilateral with

four sides equal in

length and four right

angles.

18.
Given two line segments l and m on a piece of paper, we can fold the paper

about the mid point of l in such a way that l folds onto itself.

Then m is parallel to l if and only if m also folds onto itself or an

extension of itself.

fol

d lin

e

m

l

about the mid point of l in such a way that l folds onto itself.

Then m is parallel to l if and only if m also folds onto itself or an

extension of itself.

fol

d lin

e

m

l

19.
A diagonal of a polygon is a line segment that connects two non-

adjacent vertices of that polygon.

There are always two diagonals in a quadrilateral;

but they may not be of the same length; and they

may not be perpendicular to each other.

adjacent vertices of that polygon.

There are always two diagonals in a quadrilateral;

but they may not be of the same length; and they

may not be perpendicular to each other.

20.
Let P be the point of intersection of l and m. Fold the paper at point P

such that l folds onto itself.

Then l and m are perpendicular if an only if m is on the fold line

fold line

l

m

such that l folds onto itself.

Then l and m are perpendicular if an only if m is on the fold line

fold line

l

m

21.
Section 12.3

Relationships between Geometric Shapes– Level 2

Students at this level will start to notice the relationships among

different types of triangles.

Triangles

Scalene triangles Isosceles triangles

Equilateral triangles

Right triangles

Relationships between Geometric Shapes– Level 2

Students at this level will start to notice the relationships among

different types of triangles.

Triangles

Scalene triangles Isosceles triangles

Equilateral triangles

Right triangles

22.
More Names for quadrilaterals

Model Abstraction Description

Kite Kite Quadrilateral with

two non-overlapping

pairs of adjacent

sides with the same

length

Tail of airplane Trapezoid Quadrilateral with

exactly one pair of

parallel sides

Table/desk Isosceles trapezoid Quadrilateral with

exactly one pair of

parallel sides and the

remaining sides are

of the same length

Model Abstraction Description

Kite Kite Quadrilateral with

two non-overlapping

pairs of adjacent

sides with the same

length

Tail of airplane Trapezoid Quadrilateral with

exactly one pair of

parallel sides

Table/desk Isosceles trapezoid Quadrilateral with

exactly one pair of

parallel sides and the

remaining sides are

of the same length

23.
Summary

Quadrilateral

Trapezoid Parallelogram Kite

Isosceles Rectangle Rhombus

Square

Quadrilateral

Trapezoid Parallelogram Kite

Isosceles Rectangle Rhombus

Square

24.
parallelograms

rectangles

kites

rhombi

squares

trapezoids

Isosceles

trapezoids

rectangles

kites

rhombi

squares

trapezoids

Isosceles

trapezoids

25.
Section 12.4

An Introduction to a Formal Approach to Geometry

- Level 3

In this section, we will learn more technical terms for lines and angles.

We will also introduce the measure of an angle for the purpose of

formally defining right angles, perpendicular lines, acute and obtuse

triangles.

In the end, we will also see the different types of angles associate with

parallel lines.

An Introduction to a Formal Approach to Geometry

- Level 3

In this section, we will learn more technical terms for lines and angles.

We will also introduce the measure of an angle for the purpose of

formally defining right angles, perpendicular lines, acute and obtuse

triangles.

In the end, we will also see the different types of angles associate with

parallel lines.

26.
Points and Lines in a Plane

In formal geometry, the notions of a point and a line are usually left

undefined (a so-called primitive object). Their properties are then

determined by the axioms which refer to them.

Some terminologies

a line collinear points

concurrent lines non-concurrent

lines

In formal geometry, the notions of a point and a line are usually left

undefined (a so-called primitive object). Their properties are then

determined by the axioms which refer to them.

Some terminologies

a line collinear points

concurrent lines non-concurrent

lines

27.
Two different given lines L1 and L2 on a plane are said to be parallel if

they will never intersect each other no matter how far they are

Any line L is also said to be parallel to itself.

they will never intersect each other no matter how far they are

Any line L is also said to be parallel to itself.

28.

29.
The ray CD is defined to be the set of all

points of the line CD that are on the same

side of C as point D.

An angle is the union of two rays with a common endpoint.

i d e

s

vertex interior

side

points of the line CD that are on the same

side of C as point D.

An angle is the union of two rays with a common endpoint.

i d e

s

vertex interior

side

30.
Any angle divides the plane into 3 non-overlapping parts:

a)The angle itself (two rays with the same end point),

b)The interior of the angle,

c)The exterior of the angle.

The interior of an angle is defined to the region R in the plane

with the property that:

“the segment connecting any two points in this region R is

also completely contained in R.”

This property is called the Convex property. If a region is not convex,

it will be called concave.

a)The angle itself (two rays with the same end point),

b)The interior of the angle,

c)The exterior of the angle.

The interior of an angle is defined to the region R in the plane

with the property that:

“the segment connecting any two points in this region R is

also completely contained in R.”

This property is called the Convex property. If a region is not convex,

it will be called concave.

31.
Convex and Concave Shapes

A figure is convex if a line segment joining any two points inside the

figure lies completely inside the figure.

A figure is convex if a line segment joining any two points inside the

figure lies completely inside the figure.

32.
More Examples

Category 1 Category 2

What is the mathematical property that separates these two categories of shapes?

Answer: Convex property or Concave property.

Category 1 Category 2

What is the mathematical property that separates these two categories of shapes?

Answer: Convex property or Concave property.

33.
Angles are measured by a semi-circular device

called a protractor.

The whole circle is divided into 360 equal parts, each

part is defined to have measure one degree (written 1°).

Hence a semi-circular protractor has 180 degrees.

One degree is divided into 60 minutes and one minute is further

divided into 60 seconds.

27 degrees 35 minutes 41 seconds is written as 27°35’41”

Two angles are said to be congruent if they have the same measure.

called a protractor.

The whole circle is divided into 360 equal parts, each

part is defined to have measure one degree (written 1°).

Hence a semi-circular protractor has 180 degrees.

One degree is divided into 60 minutes and one minute is further

divided into 60 seconds.

27 degrees 35 minutes 41 seconds is written as 27°35’41”

Two angles are said to be congruent if they have the same measure.

34.
Names of angles

A straight angle has 180 degrees. (this type of angle has

no interior nor exterior)

An obtuse angle has measure between 90° and 180°.

A right angle has exactly 90°.

An acute angle has measure less than 90°.

A straight angle has 180 degrees. (this type of angle has

no interior nor exterior)

An obtuse angle has measure between 90° and 180°.

A right angle has exactly 90°.

An acute angle has measure less than 90°.

35.
Classification of triangles according to their angles.

A triangle with one right angle is called a right triangle.

A triangle with one obtuse angle is called an obtuse triangle.

A triangle with 3 acute angles is called an acute triangle.

A triangle with one right angle is called a right triangle.

A triangle with one obtuse angle is called an obtuse triangle.

A triangle with 3 acute angles is called an acute triangle.

36.
Classification of triangles according to their sides.

A triangle with 3

A triangle with 3 different sides is called a

equal sides is called scalene triangle.

an equilateral

triangle.

A triangle with 2 equal

sides is called an

isosceles triangle.

A triangle with 3

A triangle with 3 different sides is called a

equal sides is called scalene triangle.

an equilateral

triangle.

A triangle with 2 equal

sides is called an

isosceles triangle.

37.
Two angles are called vertical angles if they are opposite to each other

and are formed by a pair of intersecting lines.

A B

Any pair of vertical angles are always congruent.

and are formed by a pair of intersecting lines.

A B

Any pair of vertical angles are always congruent.

38.
More special angles

Two angles are said to be supplementary if their measures add up to 180°.

α β

Two angles are said to be complementary if their measures add up to 90°.

α

β

Two angles are said to be supplementary if their measures add up to 180°.

α β

Two angles are said to be complementary if their measures add up to 90°.

α

β

39.
Perpendicular Lines

Two lines are said to be perpendicular to each other if they intersect to

form a right angle

Two lines are said to be perpendicular to each other if they intersect to

form a right angle

40.
Angles associate with Parallel Lines

Given two line L1 and L2 (not necessarily parallel) on the plane, a third line T

is called a transversal of L1 and L2 if it intersects these two lines.

L1

L2

T

Given two line L1 and L2 (not necessarily parallel) on the plane, a third line T

is called a transversal of L1 and L2 if it intersects these two lines.

L1

L2

T

41.
Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T

be a transversal.

a) a and form a pair of corresponding angles.

b) c and form a pair of corresponding angles etc.

L1

a

c

L2

T

be a transversal.

a) a and form a pair of corresponding angles.

b) c and form a pair of corresponding angles etc.

L1

a

c

L2

T

42.
Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T

be a transversal.

c) c and form a pair of alternate interior angles.

d) d and form a pair of alternate interior angles.

L1

c

d L2

T

be a transversal.

c) c and form a pair of alternate interior angles.

d) d and form a pair of alternate interior angles.

L1

c

d L2

T

43.
Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T

be a transversal.

e) a and form a pair of alternate exterior angles.

f) b and form a pair of alternate exterior angles.

L1

a

L2

T

be a transversal.

e) a and form a pair of alternate exterior angles.

f) b and form a pair of alternate exterior angles.

L1

a

L2

T

44.
Let l and m be two lines (not necessarily parallel) on the plane,

and T be a transversal.

1 and 3 are called interior angles on the same side of the

The above lines l and m are parallel if and only if the measure of any

pair of interior-angles-on-the-same-side-of-the-transversal add up to

and T be a transversal.

1 and 3 are called interior angles on the same side of the

The above lines l and m are parallel if and only if the measure of any

pair of interior-angles-on-the-same-side-of-the-transversal add up to

45.
Let L1 and L2 be two lines on the plane, and T be a transversal.

If L1 and L2 are parallel, then

a) any pair of corresponding angles are congruent,

b) any pair of alternate interior angles are congruent,

c) any pair of alternate exterior angles are congruent.

L1

L2

T

If L1 and L2 are parallel, then

a) any pair of corresponding angles are congruent,

b) any pair of alternate interior angles are congruent,

c) any pair of alternate exterior angles are congruent.

L1

L2

T

46.
Let L1 and L2 be two lines on the plane, and T be a transversal.

a) if there is a pair of congruent corresponding angles, then L1 and L2

are parallel.

b) if there is a pair of congruent alternate interior angles, then L1 and L2

are parallel.

c) if there is a pair of congruent alternate exterior angles, then L1 and L2

are parallel.

L1

L2

T

a) if there is a pair of congruent corresponding angles, then L1 and L2

are parallel.

b) if there is a pair of congruent alternate interior angles, then L1 and L2

are parallel.

c) if there is a pair of congruent alternate exterior angles, then L1 and L2

are parallel.

L1

L2

T

47.
Angle Sum in a Triangle

Draw an arbitrary triangle on a

piece of paper and label all 3

b angles.

Next cut out the triangle, and

then cut it into 3 parts (as

indicated by the dashed lines)

Arrange the 3 angles side by

a side, can you get a straight

c angle?

Conclusion:

The angle sum in a triangle is always 180°

Draw an arbitrary triangle on a

piece of paper and label all 3

b angles.

Next cut out the triangle, and

then cut it into 3 parts (as

indicated by the dashed lines)

Arrange the 3 angles side by

a side, can you get a straight

c angle?

Conclusion:

The angle sum in a triangle is always 180°

48.
Angle Sum in a Quadrilateral

If we draw a diagonal, then

the quadrilateral will be cut

into two triangles. Hence

the angle sum of a

quadrilateral must be twice

the sum of angles in a

triangle, i,e, 360.

If we draw a diagonal, then

the quadrilateral will be cut

into two triangles. Hence

the angle sum of a

quadrilateral must be twice

the sum of angles in a

triangle, i,e, 360.

49.
Application of Degree Measure

Angles can be used to indicate directions. The only difference

is that the measure can be greater than 180º.

In navigation, the direction can be any value between 0º and

Angles can be used to indicate directions. The only difference

is that the measure can be greater than 180º.

In navigation, the direction can be any value between 0º and

50.
The Bearing System

The exact (magnetic) North is defined to be 0 degree.

Any other direction is defined to be the number of

degrees away from exact North measuring in the

clockwise direction.

N

130º

south east direction

The exact (magnetic) North is defined to be 0 degree.

Any other direction is defined to be the number of

degrees away from exact North measuring in the

clockwise direction.

N

130º

south east direction

51.
The Bearing System

In particular, 90º is equal to exact East,

N

90º = East

In particular, 90º is equal to exact East,

N

90º = East

52.
The Bearing System

and 180º is equal to exact South,

N

180º = South

and 180º is equal to exact South,

N

180º = South

53.
The Bearing System

and 270º is equal to exact West,

N

270º = West

and 270º is equal to exact West,

N

270º = West

54.
Runway Numbers

In any airport, each runway is assigned a number

according to the direction it is pointing at – except

that the units digit is omitted for simplicity.

For example, runway 24 is actually pointing at 240º,

and it means that during final approach, the aircraft is

heading 240º - which is about south west.

In any airport, each runway is assigned a number

according to the direction it is pointing at – except

that the units digit is omitted for simplicity.

For example, runway 24 is actually pointing at 240º,

and it means that during final approach, the aircraft is

heading 240º - which is about south west.

55.
This is one of the many signs that you will see in a big commercial airport. It

tells the pilots which runway is in front of them.

tells the pilots which runway is in front of them.

56.
San Diego International

57.
Montgomery Field

58.
Section 12.5

Regular Polygons, Tessellations,

and Circles

Regular Polygons, Tessellations,

and Circles

59.
Basic Definitions

A simple closed curve in the plane is a curve that can be traced with the

same starting and stopping points and without crossing or retracing any

part of the curve.

A simple closed “curve” that is made up with line segments is called a

A polygon where all sides are congruent and all angles are congruent is

called a regular polygon.

A simple closed curve in the plane is a curve that can be traced with the

same starting and stopping points and without crossing or retracing any

part of the curve.

A simple closed “curve” that is made up with line segments is called a

A polygon where all sides are congruent and all angles are congruent is

called a regular polygon.

60.
The word "polygon" derives from the Greek poly, meaning "many",

and gonia, meaning "angle".

Equilateral Square Regular

triangle n=4 pentagon

n=3 n=5

Regular Regular Regular

hexagon heptagon octagon

n=6 n=7 n=8

and gonia, meaning "angle".

Equilateral Square Regular

triangle n=4 pentagon

n=3 n=5

Regular Regular Regular

hexagon heptagon octagon

n=6 n=7 n=8

61.
Polygons and their nomenclature

A Triangle (from Latin) has 3 sides

A Quadrilateral (from Latin) has 4 sides (tetra is from Greek)

A Pentagon (from Greek) has 5 sides

A Hexagon (from Latin) has 6 sides

A Heptagon (from Greek) (or a Septagon from Latin?) has 7 sides

A Triangle (from Latin) has 3 sides

A Quadrilateral (from Latin) has 4 sides (tetra is from Greek)

A Pentagon (from Greek) has 5 sides

A Hexagon (from Latin) has 6 sides

A Heptagon (from Greek) (or a Septagon from Latin?) has 7 sides

62.
In fact, “Septagon” is not an official word for the 7-gon, it is not even in a

dictionary. It was invented by some elementary school teachers to make it

easier to remember. The Latin word septem means 7 and September means

the seventh month.

The old Roman calendar began the year in January, (named after the Roman

god of fortune, Janus), and September was the seventh month. Afterwards,

Julius Augustus (46 BC) named two more then-29 day periods after himself

and September came to be as we know it in the Gregorian Calendar, the ninth

dictionary. It was invented by some elementary school teachers to make it

easier to remember. The Latin word septem means 7 and September means

the seventh month.

The old Roman calendar began the year in January, (named after the Roman

god of fortune, Janus), and September was the seventh month. Afterwards,

Julius Augustus (46 BC) named two more then-29 day periods after himself

and September came to be as we know it in the Gregorian Calendar, the ninth

63.
An Octagon (from Greek) has 8 sides

A Nonagon (from Latin) has 9 sides.

A Decagon (from Greek) has 10 sides.

A polygon with more than n (>10) sides is usually just

called an n-gon.

A Nonagon (from Latin) has 9 sides.

A Decagon (from Greek) has 10 sides.

A polygon with more than n (>10) sides is usually just

called an n-gon.

64.
Click to scroll up.

65.
Angle Sum in other Polygons

What is the sum of all angles in a quadrilateral?

Answer: 180 × 2 = 360

What is the sum of all angles in a pentagon?

Answer: 180 × 3 = 540

What is the sum of all angles in a quadrilateral?

Answer: 180 × 2 = 360

What is the sum of all angles in a pentagon?

Answer: 180 × 3 = 540

66.
Angle Sum in other Polygons

For a polygon with n sides, the angle sum is

(n – 2) × 180°

For a polygon with n sides, the angle sum is

(n – 2) × 180°

67.
Angles in a polygon

In a regular pentagon:

the measure of a central angle is 360°/5 = 72°

the measure of an exterior angle is also 360°/5 = 72°

the measure of a vertex angle is 180° – 72° = 108°

In a regular pentagon:

the measure of a central angle is 360°/5 = 72°

the measure of an exterior angle is also 360°/5 = 72°

the measure of a vertex angle is 180° – 72° = 108°

68.
Angles & Angle Sums in Regular

polygons

For a regular pentagon,

360

m(central angle) = 5 72

le

ng

la

ra

nt

ce

m(vertex angle)

center = (3 × 180) ÷ 5

= 108

vertex angle

polygons

For a regular pentagon,

360

m(central angle) = 5 72

le

ng

la

ra

nt

ce

m(vertex angle)

center = (3 × 180) ÷ 5

= 108

vertex angle

69.
Tessellations (or Tilings)

A tessellation is an arrangement of congruent shapes that cover an

entire area with no overlaps or gaps.

A 2D geometric figure R is said to tessellate (or tile) the plane if the entire

plane can be completely covered by (an infinite number of) congruent

copies of R with no overlaps or gaps.

A tessellation is an arrangement of congruent shapes that cover an

entire area with no overlaps or gaps.

A 2D geometric figure R is said to tessellate (or tile) the plane if the entire

plane can be completely covered by (an infinite number of) congruent

copies of R with no overlaps or gaps.

70.
A regular tessellation means a tessellation made up of congruent

regular polygons.

A tessellation with congruent copies of several different regular polygons

are called semiregular tessellations

regular polygons.

A tessellation with congruent copies of several different regular polygons

are called semiregular tessellations

71.
It is also possible to tile a plane with congruent copies of several different

irregular polygons, such as below.

irregular polygons, such as below.

72.
Convex and Concave Polygons

a convex quadrilateral a concave (i.e. non-

convex) quadrilateral

A polygon X is said to be convex if you take any two points on X (including

the boundary), the line segment joining them lies entirely within the tile

(again including the boundary).

a convex quadrilateral a concave (i.e. non-

convex) quadrilateral

A polygon X is said to be convex if you take any two points on X (including

the boundary), the line segment joining them lies entirely within the tile

(again including the boundary).

73.
Question: What kind of polygons can tessellate the

1. Can triangles tessellate the plane?

Any triangle can tessellate the plane.

2. Can quadrilaterals tessellate the plane?

Any square can tessellate the plane.

Any rectangle can tessellate the plane.

Any convex quadrilateral can tessellate the plane.

In fact, any quadrilateral (including non-convex ones) can

tessellate the plane.

3. Can regular pentagons tessellate the plane?

No, a regular pentagon will not tessellate the plane.

4. Can hexagons tessellate the plane?

Any regular hexagon can tessellate the plane.

In fact, exactly 3 classes of convex hexagons can tile the plane.

(this was proved by K. Reinhardt in his 1918 doctoral thesis.

See next slide.)

1. Can triangles tessellate the plane?

Any triangle can tessellate the plane.

2. Can quadrilaterals tessellate the plane?

Any square can tessellate the plane.

Any rectangle can tessellate the plane.

Any convex quadrilateral can tessellate the plane.

In fact, any quadrilateral (including non-convex ones) can

tessellate the plane.

3. Can regular pentagons tessellate the plane?

No, a regular pentagon will not tessellate the plane.

4. Can hexagons tessellate the plane?

Any regular hexagon can tessellate the plane.

In fact, exactly 3 classes of convex hexagons can tile the plane.

(this was proved by K. Reinhardt in his 1918 doctoral thesis.

See next slide.)

74.
Type 2 Type 3

Type 1

A + B + D = 360° A = C = E = 120°

B + C + D = 360°

C + E + F = 360° a = a'

A + E + F = 360°

a=d c = c'

c=e e = e'

Regular hexagon

belongs to Type 3

Type 1

A + B + D = 360° A = C = E = 120°

B + C + D = 360°

C + E + F = 360° a = a'

A + E + F = 360°

a=d c = c'

c=e e = e'

Regular hexagon

belongs to Type 3

75.
8. He then went on to explore the tessellations by irregular but convex

pentagons and found 5 classes that do tile the plane.

He felt that he had found all of them even though he could not give

a proof because he claimed that it would be very tedious to do so.)

pentagons and found 5 classes that do tile the plane.

He felt that he had found all of them even though he could not give

a proof because he claimed that it would be very tedious to do so.)

76.
In 1968, after 35 years working on the problem on and off, R. B. Kershner,

a physicist at Johns Hopkins University, discovered 3 more classes of

pentagons that will tessellate.

Kershner was sure that he had found all of them, but again did not offer a

complete proof, which “would require a rather big book.”

Shortly after an article of the “complete” classification of convex pentagons

into 8 types appeared in Scientific American (July 1975), an amateur

mathematician (R. James III) discovered a 9th type!

Between 1976 and 1977, a San Diego housewife Marjorie Rice, without

formal education in mathematics beyond high school, found 4 more types!

A 14th type was found by a mathematics graduate student in 1985.

In 2015, University of Washington Bothell mathematicians Casey Mann,

Jennifer McLoud, and David Von Derau discovered a 15th regular tiling

with convex pentagon using a computer algorithm.

Yet, no one can say that this is the complete list.

(the 15 types of pentagons that tile the plane)

a physicist at Johns Hopkins University, discovered 3 more classes of

pentagons that will tessellate.

Kershner was sure that he had found all of them, but again did not offer a

complete proof, which “would require a rather big book.”

Shortly after an article of the “complete” classification of convex pentagons

into 8 types appeared in Scientific American (July 1975), an amateur

mathematician (R. James III) discovered a 9th type!

Between 1976 and 1977, a San Diego housewife Marjorie Rice, without

formal education in mathematics beyond high school, found 4 more types!

A 14th type was found by a mathematics graduate student in 1985.

In 2015, University of Washington Bothell mathematicians Casey Mann,

Jennifer McLoud, and David Von Derau discovered a 15th regular tiling

with convex pentagon using a computer algorithm.

Yet, no one can say that this is the complete list.

(the 15 types of pentagons that tile the plane)

77.
The elevator lobby on the 7th floor

of the Mathematics building in the

Ohio State University.

Note that the tiles are irregular

pentagons, and yet they tessellate

the plane.

of the Mathematics building in the

Ohio State University.

Note that the tiles are irregular

pentagons, and yet they tessellate

the plane.

78.
Columnar Basalts -

Naturally occurring hexagonal columns

that tessellate the plane.

Naturally occurring hexagonal columns

that tessellate the plane.

79.
With the situation so intricate for convex pentagons, you might think that it

must be even worse for polygons with 7 or more sides. However, the

situation is remarkably simple, as Reinhardt proved in 1927:

A convex polygon with 7 or more sides cannot tessellate.

must be even worse for polygons with 7 or more sides. However, the

situation is remarkably simple, as Reinhardt proved in 1927:

A convex polygon with 7 or more sides cannot tessellate.

80.
Circles

A Circle is the set of all points in the plane that are at a fixed distance

from a given point called the center.

The distance from any point on the circle to the center is called the

radius of the circle. Any segment connect the center to the edge is also

called a radius.

The length of any line segment whose endpoints are on the circle and

which contains the center is called the diameter of the circle. The

segment is also called a diameter of the circle.

A Circle is the set of all points in the plane that are at a fixed distance

from a given point called the center.

The distance from any point on the circle to the center is called the

radius of the circle. Any segment connect the center to the edge is also

called a radius.

The length of any line segment whose endpoints are on the circle and

which contains the center is called the diameter of the circle. The

segment is also called a diameter of the circle.

81.
Circles

Circles have the following 3 properties that make them very useful.

1. They are highly symmetrical, hence they have a sense of beauty and

are often used in designs due to aesthetical reasons.

eg. dinnerware.

However, plates

don’t have to be

Circles have the following 3 properties that make them very useful.

1. They are highly symmetrical, hence they have a sense of beauty and

are often used in designs due to aesthetical reasons.

eg. dinnerware.

However, plates

don’t have to be

82.

83.
Circles

Circles have the following 3 properties that make them very useful.

1. They are high symmetrical, hence they have a sense of beauty and

are often used in designs.

eg. dinnerware.

However, plates

don’t have to be

Circles have the following 3 properties that make them very useful.

1. They are high symmetrical, hence they have a sense of beauty and

are often used in designs.

eg. dinnerware.

However, plates

don’t have to be

84.
2. A circle has a center, and every point on a circle bears the same distance

from the center. This is called the constant radius property.

Applications: wheels and gears.

Note: An ellipse also has a center, but the radius varies in length

according to its placement.

from the center. This is called the constant radius property.

Applications: wheels and gears.

Note: An ellipse also has a center, but the radius varies in length

according to its placement.

85.
3. For a given (fixed) perimeter, the circle has the largest area.

Applications: water bottles, soda cans, and any container for

pressurized liquid are all cylindrical in shape.

Applications: water bottles, soda cans, and any container for

pressurized liquid are all cylindrical in shape.

86.

87.
Section 12.6

Describing 3-Dimensional Shapes

(This will be taught after section 13.2)

Describing 3-Dimensional Shapes

(This will be taught after section 13.2)