Polygon Geometry: Pentagons, Hexagons and Dodecagons

Dodecagon-shaped Jamaican One Cent Coin
Dodecagon-shaped Jamaican One Cent Coin. De Agostini / A. Dagli Orti/Getty Images
01
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What is a Polygon?

Dodecagon-shaped Jamaican One Cent Coin
Dodecagon-shaped Jamaican One Cent Coin. De Agostini / A. Dagli Orti/Getty Images

Polygons are Two Dimensional

In geometry, a polygon is any two-dimensional shape that:

  • Is made up of three or more straight lines
  • Is closed – there are no openings or breaks in the shape
  • Has pairs of lines that connect at the corners or vertices where they form angles
  • Has an equal number of sides and interior angles (the angles inside the polygon)

(Two-dimensional means flat – like a piece of paper)

It's All Greek

The name polygon comes from two Greek words:

  • poly- meaning "many"
  • gon meaning "angle"

Shapes That Are Polygons

  • Triangles
  • Squares
  • Pentagons
  • Octagons
  • Dodecagons - such as the 12-sided Jamaican coin in the image above

Shapes That Are Not Polygons

  • A circle - because it is composed of a curved line rather than straight lines
  • A box or cube - because it is three-dimensional in shape not two-dimensional
02
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Naming Polygons

Common Polygons and Their Internal Angles
Common Polygons From Triangles to Decagons. © Ted French

Polygon Names

The names of individual polygons are derived from the number of sides and/or interior angles the shape possesses.

(By the way, the number of interior angles - angles inside the shape - must always equal the number of sides).

The common names of most polygons have the Greek prefix for the number of angles attached to the Greek word for angle (gon).

So, the common names for five and six-sided regular polygons are:

  • penta (Greek meaning five) + gon pentagon
  • hexa (Greek meaning six) + gon hexagon

The Exceptions

There are, of course, exceptions to this naming scheme. Most notably:

Triangle- uses the Greek prefix Tri, but instead of the Greek gon, the Latin angle is used. (Rarely are they called trigons).

Quadrilateral - is derived from the Latin prefix quadri - meaning four - attached to the word lateral - which is another Latin word meaning side. 

Sometimes, a four-sided polygon is referred to as a quadrangle or tetragon.

n-gons

Polygons with more than ten sides and angles exist, and some have common names - such as the 100 sided hectogon.

Since they are encountered infrequently, however, they are more often given a name that attaches the number of sides and angles to the general term for angle - gon.

So, a 100-sided polygon is usually referred to as a 100-gon.

Some other n-gons and common names for polygons with more than ten sides are:

  • 11-gon: Hendecagon
  • 12-gon: Dodecagon
  • 20-gon: Icosagon
  • 50-gon: Pentecontagon
  • 1000-gon: Chiliagon
  • 1000000-gon: Megagon

Polygon Limit

Theoretically, there is no limit to the number of sides and angles for a polygon.

As the size of the interior angles of a polygon get smaller and the length of its sides get shorter a polygon approaches a circle - but it never quite gets there.

03
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Classifying Polygons

Regular, Irregular, Complex, Simple Hexagons
Different Types of Hexagons/Hexagam. © Ted French

Regular vs. Irregular Polygons

In a regular polygon all of the angles are of equal size and all of the sides are equal in length.

An irregular polygon is any polygon that does not have equal sized angles and sides of equal length.

Convex vs. Concave

A second way to classify polygons is by the size of their internal angles. The two choices are convex and concave:

Simple vs. Complex Polygons

Yet another way to classify polygons is by the way the lines forming the polygon intersect.

  • The lines of simple polygons connect or intersect only once - at the vertices;
  • The lines of complex polygons intersect more than once.

The names of complex polygons are sometimes different from those of simple polygons with the same number of sides.

For example,

  • A regular-shaped hexagon is a six-sided simple polygon
  • A star-shaped hexagram is a six-sided complex polygon created by overlapping two equilateral triangles as shown in the image above
04
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Sum of the Interior Angles Rule

Calculating the Internal Angles of a Polygon
Calculating the Internal Angles of a Polygon. Ian Lishman/Getty Images

As a rule, each time a side is added to a polygon, such as:

  • From a triangle to quadrilateral (three to four sides)
  • From a pentagon to a hexagon (five to six sides)

another 180° is added to the sum total of the interior angles.

This rule can be written as a formula:

(n - 2) × 180°

where n = number of sides of the polygon.

So the sum of the interior angles for a hexagon can be found by using the formula:

(6 - 2) × 180° = 720°

How Many Triangles in that Polygon?

The above interior angle formula is derived by dividing a polygon up into triangles, and this number can be found with the calculation:

 n - 2

where n again is equal to the number of sides of the polygon.

So, a hexagon (six sides) can be divided into four triangles (6 - 2) and a dodecagon into 10 triangles (12 - 2).

Angle Size for Regular Polygons

For regular polygons (angles all the same size and sides all the same length), the size of each angle in a polygon can be calculated by dividing the total number of degrees by the total number of sides.

For a regular six-sided hexagon, each angle is:

720° ÷ 6 = 120°

05
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Some Well-Known Polygons

The Octagon - A Regular Eight Sided Octagon
The Octagon - A Regular Eight Sided Octagon. Scott Cunningham/Getty Images

Triangular Trusses

Roof trusses - are often triangular in shape. Depending on the width and pitch of the roof, the truss might incorporate equilateral and isosceles triangles.

Because of their great strength, triangles are also used in the construction of bridges, bicycle frames, and the Eiffel Tower.

The Pentagon

The Pentagon – the headquarters for the U.S. Department of Defense – takes its name from its shape. It is a five-sided regular pentagon.

Home Plate

Another well-known five-sided regular pentagon is home plate on a baseball diamond.

The Fake Pentagon

A giant shopping mall near Shanghai, China is built in the shape of a regular pentagon, and is sometimes called the Fake Pentagon because of its resemblance to the original.

Snowflakes

Every snowflake starts out as a hexagonal plate, but temperature and moisture levels add branches and tendrils so that each one ends up looking different..

Bees and Wasps

Natural hexagons also include beehives where each cell in a honeycomb that the bees construct to hold honey is hexagonal in shape.

The nests of paper wasps also contain hexagonal cells where they raise their young.

The Giant's Causeway

Hexagons are also found in the Giant's Causeway located in north-east Ireland.

It is a natural rock formation composed of about 40,000 interlocking basalt columns that were created as the lava from an ancient volcanic eruption slowly cooled.

The Octagon

The Octagon pictured above – the name given to the ring or cage used in UFC (Ultimate Fighting Championship) bouts – takes its name from its shape. It is an eight-sided regular octagon.

Stop Signs

The stop sign – one of the best known traffic signs – is another eight-sided regular octagon.

Although the color and the wording or symbols on the sign may vary, the octagonal shape for the stop sign is used in many countries around the world.