Polygon Geometry: Pentagons, Hexagons and Dodecagons

Two-dimensional regular polygons are everywhere

Few geometric shapes are as diverse as polygons. They include the familiar triangle, square, and pentagon, but that is only the start. 

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

  • Is made up of three or more straight lines
  • Is closed with 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

Two-dimensional means flat like a piece of paper. Cubes are not polygons because they are three-dimensional. Circles are not polygons because they don't contain straight lines.

About Polygons

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

The name polygon comes from two Greek words:

  • Poly, which means many
  • Gon, which means angle

Shapes That Are Polygons

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

How Polygons Are Named

Common Polygons and Their Internal Angles

Ted French

The names of individual polygons are derived from the number of sides or interior angles the shape possesses. The number of interior angles always equals 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 prefixes and common names for five- and six-sided regular polygons are:

  • Penta (Greek meaning five) + gon pentagon
  • Hexa (Greek meaning six) + gon hexagon

There are exceptions to this naming scheme. Most notably:

  • Triangle uses the Greek prefix Tri, but instead of the Greek gon, the Latin angle is used. Trigon is rarely used.
  • 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 10 sides and angles exist, and some have common names, such as the 100-sided hectogon.

They are encountered infrequently, so they are often given a name that attaches the number of sides and angles to the general term for angle, which is gon.

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

Some other n-gons and common names for polygons with more than 10 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 gets smaller, and the length of its sides gets shorter, the polygon approaches a circle, but it never quite gets there.

Classifying Polygons

Regular, Irregular, Complex, Simple Hexagons

Ted French

Regular vs. Irregular Polygons

In a regular polygon, all of the angles are of equal size, and all 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 Polygons

A second way to classify polygons is by the size of their internal angles. They are either convex or concave:

  • Convex polygons have no internal angles greater than 180°.
  • Concave polygons have at least one internal angle that is greater than 180°.

Simple vs. Complex Polygons

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.

Sum of the Interior Angles Rule

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 total of the interior angles.

This rule can be written as a formula:

(n - 2) × 180°

where n equals the 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.

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, in which angles are all the same size and sides are 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°

Some Well-Known Polygons

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

Triangular Trusses

Roof trusses are often triangular. Depending on the width and pitch of the roof, the truss might incorporate equilateral or isosceles triangles. Because of their great strength, triangles are used in the construction of bridges and bicycle frames, and they are prominent in 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.

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

Stop Signs

The stop sign — one of the most familiar 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.