Polygons

A polygon is a plane shape with straight sides.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon (straight sides)	Not a Polygon (has a curve)	Not a Polygon (open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Regular or Irregular

If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

Regular		Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.
If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a "cave" in it)

Convex		Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon (this one's a Pentagon)		Complex Polygon (also a Pentagon)

More Examples

Irregular Hexagon		Concave Octagon		Complex Polygon (a "star polygon", in this case a pentagram)

Play With Them! Try Interactive Polygons ... make them regular, concave or complex.

Names of Polygons

		If it is a Regular Polygon...
Name	Sides	Shape	Interior Angle
Triangle (or Trigon)	3		60°
Quadrilateral (or Tetragon)	4		90°
Pentagon	5		108°
Hexagon	6		120°
Heptagon (or Septagon)	7		128.571°
Octagon	8		135°
Nonagon (or Enneagon)	9		140°
Decagon	10		144°
Hendecagon (or Undecagon)	11		147.273°
Dodecagon	12		150°
Triskaidecagon	13		152.308°
Tetrakaidecagon	14		154.286°
Pentadecagon	15		156°
Hexakaidecagon	16		157.5°
Heptadecagon	17		158.824°
Octakaidecagon	18		160°
Enneadecagon	19		161.053°
Icosagon	20		162°
Triacontagon	30		168°
Tetracontagon	40		171°
Pentacontagon	50		172.8°
Hexacontagon	60		174°
Heptacontagon	70		174.857°
Octacontagon	80		175.5°
Enneacontagon	90		176°
Hectagon	100		176.4°
Chiliagon	1,000		179.64°
Myriagon	10,000		179.964°
Megagon	1,000,000		~180°
Googolgon	10100		~180°
n-gon	n		(n-2) × 180° / n

Types of Triangles

Triangles can be classified by various properties relating to their angles and sides. The most common classifications are described on this page.

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Classifications of Triangles

Right Triangles
 

A right triangle has one 90° and a variety of often-studied properties including: Right Triangle Calculator Proof of Pythagorean Theorem Pythagorean Triplets Sine, Cosine, Tangent Pictures of Right Triangles 7 24 25 Right Triangle Images 3 4 5 Right Triangles 5 12 13 Right Triangles

The Equilateral triangle shown on the left has three equal sides and three equal angles.
Each angle is 60°

The Isosceles triangle shown on the left has two equal sides and two equal angles.

The Scalene Triangle has no congruent sides. In other words, each side must have a different length..

The Acute Triangle has three acute angles (an acute angle measures less than 90°)

The  Obtuse Triangle has an obtuse angle (an obtuse angle has more than 90°). In the picture on the left, the shaded angle is the obtuse angle that distinguishes this triangle

Since the total degrees in any triangle is 180°, an obtuse triangle can only have one angle that measures more than 90°.

Points, lines and planes

The word ‘geometry’ derives from the Greek words for ‘earth’ (geo) and ‘to measure’ (metron).

A construction is a geometric drawing for which only a compass and a straightedge may be used.

a compass: for making circles; for transferring the distance between two existing points

ruler/straightedge: used for drawing a line between two existing points; no distance markings will be used; no measurements are allowed

A conjecture is an educated guess.
Using specific observations and examples to arrive at a conjecture is called inductive reasoning.
For example, you might make the conjecture that for all real numbers x and y, the distance between them is given by the formula y−x.
A counterexample is a specific example that shows that a conjecture is not always true.
For example, here is a counterexample to the previous conjecture:
Let y=5 and x=7. Then, y−x=5−7=−2, which is not the distance between them.
Deductive reasoning uses logic, and statements that are already accepted to be true, to reach conclusions.
The methods of mathematical proof are based on deductive reasoning.
Point, line, and plane are three undefined terms to get us started in the study of geometry—we will just agree on their meaning.
A point represents an exact location. It is represented with a dot. Capital letters, like P, are frequently used to denote points.

DEFINITIONS space; geometric figure

Space is the set of all points.

A geometric figure is a subset of space.

That is, a geometric figure is any collection of points.
Of course, there are certain important geometric figures (like triangles and circles) that will be studied throughout the course.
A line has length only; it has no width or thickness; it extends forever in both directions.
A line will be denoted using a lowercase script letter, like ℓ.
If A and B are two distinct points, then they determine a unique line which will be denoted by AB←→ or BA←→.


A plane is a flat surface that extends infinitely in all directions; it has length and width only; it has no thickness.
A plane will be denoted using an uppercase script letter, like P.
If A, B, and C are three distinct noncollinear points (see below), then they determine a unique plane which will be denoted by ABC.


Note: In the following definitions, the prefix ‘co’ means ‘same’.

DEFINITIONS collinear/noncollinear/coplanar/noncoplanar points

collinear points: three or more points lying on the same line


noncollinear points: points not lying on the same line
coplanar points: points lying in the same plane
noncoplanar points: points not lying in the same plane

Coplanar can also refer to other geometric figures.
For example, two lines are coplanar if and only if they lie in the same plane.