What is a Polygon?
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
Naming Polygons
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.
Classifying Polygons
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:
- Convex polygons - no internal angles greater than 180°;
- Concave polygons - at least one internal angle is greater than 180°.
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
Sum of the Interior Angles Rule
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°
Some Well-Known Polygons
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.