# What are polygons?

Can you tell your hexagons from your heptagons? Your octagons from your decagons? We explain how polygons are taught in primary school, with examples of regular and irregular versions of each multi-sided shape. Login or Register to add to your saved resources

A polygon is a flat, two-dimensional (2D) shape with straight sides that is fully closed (all the sides are joined up). The sides must be straight. Polygons may have any number of sides.

 A polygon A shape with curved sides is not a polygon. A shape that is not fully closed is not a polygon.

## Regular and irregular polygons

A regular polygon is a polygon in which all sides are of all the same length and at the same angles.

An irregular polygon is a polygon with sides and/or angles of differing lengths and sizes. (Although they still must be straight and joined up.)

### Primary-school polygons

Polygons taught in primary school include:

 Regular Irregular Triangle(3 sides) Quadrilateral(4 sides) Pentagon(5 sides) Hexagon(6 sides) Heptagon(7 sides) Octagon (8 sides) Nonagon (9 sides) Decagon(10 sides)

*A regular quadrilateral has a special name. It is called a square.

*There are different types of triangles: an equilateral triangle is a regular triangle and a right-angle triangle and an isosceles triangle are examples of irregular triangles.

### When and how polygons are taught in primary school

 What is taught How it might be taught Year 1 Children are taught to name common 2D shapes including polygons such as squares, rectangles, triangles, pentagons, hexagons and octagons. Children may learn the shapes through matching activities, going on a shape walk in the school grounds, flash-cards and games. They will look at real-life examples of shapes as well as pictures. Year 2 Children will be taught to identify properties of shapes such as the number of sides and vertices (corners). Children will count the number of sides and corners on the shape. They will describe shapes using the properties, for example: This shape has 3 corners and 3 sides. What is the shape? Year 3 Children will extend their knowledge of polygons to include different types of triangles and quadrilaterals. They will be introduced to heptagons, nonagons and decagons. Knowledge of shape properties will include angles and symmetry of these polygons. Children will describe shapes and identify them using their properties including symmetry and angles. They might be asked to sort shapes according to their properties using Venn diagrams and Carroll diagrams. Year 4 Children are taught to compare lengths and angles of polygons to decide if they are regular or irregular. The vocabulary ‘polygon’, ‘regular’ and ‘irregular’ will be used. Children will be given a range of polygons to sort into regular and irregular; this might be be completing practical tasks or using ICT. Year 5

## What is a Polygon?: Definition, Types, Formulas, Videos, Solved Examples

Have you heard about a polygon? What exactly comes to your mind when you think of a polygon? The tiles on which you walk are probably square or may be hexagonal which definitely means its a polygonal. From real-life objects, a STOP sign on the road side, a starfish or ball are all forms of a polygon. So let us try to find out what is a polygon.

### What is a Polygon?

In simple mathematics, a polygon can be any 2-dimensional shape that is formed with straight lines. Be it quadrilaterals or triangles and pentagons, these are all perfect examples of polygons. The interesting aspect is that the name of a polygon highlights the number of sides it possesses.

For example, a triangle has three sides, and a quadrilateral has four sides. So, any shape that can be drawn by connecting three straight lines is called a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral.

### Types of Polygons

We know what is a polygon. But is there more to it? Yes! Of course, there is! The polygons are categorized into different types depending on the number of sides together with the extent of the angles.

Some of the prime categories of polygons include regular polygons, irregular polygons, concave polygons, convex polygons, quadrilateral polygons, pentagon polygons and so on. Some of the most well-known polygons are triangles, squares, rectangles, parallelograms, pentagons, rhombuses, hexagons etc.

### Regular polygon

Considering a regular polygon, it is noted that all sides of the polygon are equal. Furthermore, all the interior angles remain equivalent.

### Irregular polygon

These are those polygons that aren’t regular. Be it the sides or the angles, nothing is equal as compared to a regular polygon.

### Concave polygon

A concave polygon is that under which at least one angle is recorded more than 180 degrees. Also, the vertices of a concave polygon are both inwards and outwards.

### Convex polygon

The measure of the interior angle stays less than 180 degrees for a convex polygon. Such a polygon is the exact opposite of a concave polygon. Moreover, the vertices associated to a convex polygon are always outwards.

Four-sided polygon or quadrilateral polygon is quite common. There are different versions of a quadrilateral polygon such as square, parallelogram and rectangle.

### Pentagon polygon

Pentagon polygons are six-sided polygons. It is important to note that, the five sides of the polygon stay equal in length. A regular pentagon is a prime type of pentagon polygon.

### Formulas

• Area of a regular polygon = (1/2) N sin(360°/N) S2
• where N is sides and S is the length from the centre to a corner.
• Sum of the interior angles of a polygon = (N – 2) × 180°
• The angle formed by two adjacent sides of the polygon are interior angles while angle formed by two adjacent sides outside the polygon are exterior angles.

### Solved Examples For You

Question 1: I’m going to place a rope around the perimeter of our school playground that is in the shape of an octagon. The sides are 10m, 10m, 8m, 8m, 5m, 5m, 9m, 9m. How many metres of rope will be needed for the perimeter?

Length of the rope required = perimeter of the school playground. The perimeter is the sum of all the sides of the polygon. Here the school playground is in the form of an octagon with the side as  10m, 10m, 8m, 8m, 5m, 5m, 9m, 9m

1. Perimeter =  10 + 10+ 8+ 8+ 5+ 5+ 9+ 9 = 64m
2. Length of the rope required = 64m
3. Question 2: Give a simple definition of polygon?

Answer: A polygon refers to any two-dimensional shape whose formation takes place with straight lines. Triangles, hexagons, pentagons, and quadrilaterals are all examples of polygons. The name gives an indication of how many sides the shape has. For example, a triangle consists of three sides while a quadrilateral consists of four sides.

• Question 3: Name the various types of polygons?
• Answer: The various types of polygons are pentagon, hexagon, triangle, quadrilateral, octagon, heptagon, nonagon, decagon, and dodecagon.
• Question 4: Can we say that a star is a polygon?

Answer: Yes, the star is certainly a polygon. A star, in the field of geometry, refers to a special type of polygon known as a star polygon.

Question 5: Explain what is the simplest polygon?

Answer: A simple polygon refers to a polygon whose intersection does not take place by itself and it has no holes.

A simple polygon has a flat shape that involves straight, non-intersecting line segments or “sides”. The joining of these segments or sides takes place pairwise to make a single closed path.

In such a polygon, the number of verticals always equals the number of edges.

## What are Polygons?

In today’s lesson, we’ll ask what are polygons in geometry. We’ll also discuss types of polygons – regular, simple, complex, and more.

### Definition of Polygons

So what are polygons? A polygon is a two-dimensional (flat) shape. We create polygons by connecting a number of straight lines to form a closed shape. Circles and ellipses are not polygons, because their edges are not straight lines.

We name polygons according to the number of sides they have. That is, the number of straight-line segments that are used in drawing the shape. A polygon with three sides is a triangle. A polygon with four sides is a quadrangle, and so on.

### What are Regular polygons?

If all the sides of a polygon are of equal length, and all its angles have the same measure, we call it ‘regular’.

### What are Simple polygons?

Simple polygons have sides that do not intersect or cross each other:

### What are Complex polygons?

Complex polygons have sides that intersect, for example:

### What are Convex polygons?

Convex polygons are simple polygons where all the interior angles measure less than 180°. The polygons vertices point “outward” from its center.

### What are Concave polygons?

In Concave polygons, there is at least one interior angle which measure more than 180°:

### Our focus: simple convex polygons

So now that we’ve answered the question “what are polygons,” we’ll clarify that with a few exceptions, we will only concern ourselves with simple convex polygons. It is rare to see high school geometry problems that involve concave polygons, and even rarer to see ones involving self-intersecting polygons.

In addition, even those problems that do involve such polygons typically have answers that involve treating those complex polygons as a number of simple polygons side by side. For example, look at the complex polygon illustrated above. In most cases, we could simply treat it as if it was a triangle connected to a quadrilateral at one point of intersection.

## Polygon – Definition with Examples

What is a polygon?

In geometry, a polygon can be defined as a flat or plane, two-dimensional closed shape with straight sides. It does not have curved sides.

Here are a few examples of polygons.

• Polygons can be of two types:
• Regular Polygons – Polygons that have equal sides and angles are regular polygons.
• Here are a few examples of regular polygons.

Irregular Polygons – Polygons with unequal sides and angles are irregular polygons.

Here are a few examples of irregular polygons.

 Fun Facts Quadrilaterals are also polygons.
1. All Polygons are closed figures,
2. As count of sides gets bigger,
3. The names of polygons differ!

## Polygons

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”.

### Regular or Irregular

A regular polygon has all angles equal and all sides equal, 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

Name Sides Shape Interior Angle
 If it is a Regular Polygon… 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

You can make names using this method:

 Sides Start with… 20 Icosi… 30 Triaconta… 40 Tetraconta… 50 Pentaconta… 60 Hexaconta… 70 Heptaconta… 80 Octaconta… 90 Enneaconta… 100 Hecta… etc..
 Sides …end with +1 …henagon +2 …digon +3 …trigon +4 …tetragon +5 …pentagon +6 …hexagon +7 …heptagon +8 …octagon +9 …enneagon
Example: a 62-sided polygon is a Hexacontadigon

BUT, for polygons with 13 or more sides, it is OK (and easier) to write “13-gon“, “14-gon” … “100-gon”, etc.

### Remembering

A Quad Bike has 4 wheels

### Pentagon (5 Sides)

The “Pentagon” in Washington DC has 5 sides

### Hexagon (6 Sides)

Honeycomb has Hexagons

### Septagon (7 Sides)

Think Septagon is a “Seven-agon”

### Octagon (8 Sides)

An Octopus has 8 tentacles

### Nonagon (9 Sides)

Think Nonagon is a “Nine-agon”

Think Dec

## Polygons

It can seem like the study of geometry in elementary school is nothing more than learning a bunch of definitions and then classifying objects. In this part, you’ll explore some problem solving and reasoning activities that are based in geometry. But definitions are still important! So let’s start with this one.

A polygon is:

1. a plane figure
2. that is bounded by a finite number of straight line segments
3. in which each segment meets exactly two others, one at each of its endpoints.

Just as the first step in problem solving is to understand the problem, the first step in reading a mathematical definition is to understand the definition.

• Use the definition above to draw several examples of figures that are definitely polygons. (You should be able to say why your example fits the definition.)
• Draw several non-examples as well: shapes that are definitely not polygons. (You should be able to say which part of the definition fails for your non-examples.)

• The line segments that make up a polygon are called its edges and the points where they meet are called its vertices (singular: vertex).
• Because of properties (2) and (3) in the definition, the boundaries of polygons are not self-intersecting.
Not a polygon.

## What Are Polygons?

Ha!

OK, so it’s not exactly the greatest joke in the world, but geometry is not exactly a hilarious subject. Beautiful, elegant, perhaps picturesque, but not traditionally full of funny. So we can’t be too picky, can we?

The word polygon is probably one of those words you know you’ve heard before, but you’re really not sure where that was, when that was, or what it actually meant. Sure, you probably have an intuitive feeling that polygons have something to do with geometry and shapes, but what exactly?

Lucky for you, that’s exactly what we’re talking about today.

Sponsor: This episode is brought to you by NatureBox. Discover smarter snacking with a new NatureBox each month. Get your first box FREE when you go to naturebox.com/qdt.

### What Is a Polygon?

The logical way to begin our look at the mighty polygon is to first figure out what the word “polygon” means. A lot of you will be familiar with the first part of this word—poly—which derives from the Greek word for “many.” So a polygon is many somethings…but what?

Of course, that’s where the “gon” part of polygon comes into the story. In particular, “gon” is part of a Greek word that means “angle” or “corner.” So polygon means “many angles” or “many corners.” And that's exactly what geometric shapes known as polygons contain.

Polygon means “many angles” or “many corners.”