Polygon geometry: pentagons, hexagons and dodecagons

2D regular polygons everywhere

Few geometric shapes are as varied as polygons. They include the familiar triangle, square, and pentagon, but that’s just the beginning.

In geometry, a polygon is any two-dimensional shape that satisfies the following conditions:

• Consists of three or more lines

• Closed with no holes or breaks in shape

• Has pairs of lines that connect to corners or vertices where they form angles

• Has the same number of sides and interior angles

Two-dimensional means flat, like a sheet of paper. Cubes are not polygons because they are 3D. Circles are not polygons because they do not contain straight lines.

A special kind of polygon can have angles that are not all equal. In this case it is called error polygon.

About polygons

Name polygon comes from two Greek words:

• poly † which means a lot †
• gon † which means injection

Shapes that are polygons

• Triangle (triangle): 3 sides

• Tetragon (square): 4 sides

• Pentagon: 5 sides

• Hexagon: 6 sides

• Heptagon: 7 sides

• Octagons: 8 sides

• Nonagon: 9 sides

• Decagon: 10 facets

• Undecagon: 11 sides

• Dodecagons: 12 sides

What are polygons called?

The names of the individual polygons are derived from the number of sides or angles the shape has. Polygons have the same number of sides and angles.

The common name for most polygons is the Greek prefix “sides” associated with the Greek word for “angle” (gon).

Examples of this for five- and six-sided regular polygons:

• penta (Greek for five) + rut † Pentagon
• hexa (Greek for six) + rut † hexagon

There are exceptions to this naming scheme. In particular with words most commonly used for some polygons:

• Triangle † Uses Greek prefix Trio but instead of the Greek rut used Latin injection † Trigon is a valid geometric name, but is rarely used.

• quadrilateral. Derived from the Latin prefix four, meaning four is linked to the word side, which is another latin word that means: side †

• Square † Sometimes a four-sided polygon is called (square) quadrilateral or quadrilateral †

N-gons

Polygons with more than 10 sides are not common, but follow the same Greek naming convention. This is what a 100-sided polygon is called hectogon †

In mathematics, however, it is sometimes more convenient to call pentagons n-gons †

• 11 kite: gendercagon

• 12th Gon: Dodecagon

• 20-gon: Icosagon

• 50 racer: five-pointed

• 1000-gon: chiliagon

• 1000000-gon: megagon

In mathematics, n-gons and their Greek counterparts are interchangeable.

Polygon Limit

Theoretically, there is no limit to the number of sides a polygon can have.

As the interior angles of a polygon get larger and the lengths of the sides get shorter, the polygon approaches a circle but never reaches it.

Polygon classification

Regular and irregular polygons

Polygons are classified based on whether all angles or sides are equal.

• usual polygon † All angles are the same size and all sides are the same length.
• irregular polygon † Corners or sides of the same size are not the same length.

Convex vs. Concave Polygons

The second way to classify polygons is by the size of their interior angles.

• Convex Polygons † Internal angles do not exceed 180°.
• Concave polygons † At least one interior angle is greater than 180°.

Simple and complex polygons

Another way to classify polygons is how the lines that make up the polygon intersect.

• Simple Polygons : Lines connect or intersect only once, at the vertices.
• Complex Polygons : Lines cross more than once.

The names of complex polygons sometimes differ from the names of simple polygons with the same number of sides.

For instance:

• hexagon regular shape is a hexagonal simple polygon.

• star-shaped hexagram is a hexagonal complex polygon created by superimposing two equilateral triangles.

Inside angle sum rule

Typically, every time a side is added to a polygon, for example:

• Triangle to quadrilateral (three to four sides)

• Pentagon to Hexagon (five to six sides)

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

This line can be written as a formula:

(n–2) × 180°

where n is the number of sides of the polygon.

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

(6 – 2) × 180° = 720°

How many triangles are there in this polygon?

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

n – 2

In this formula, n 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).

Angular measure for regular polygons

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

For a regular hexagon, each angle is:

720° ÷ 6 = 120°

Some famous polygons

farms

Farms are often triangular in shape. Depending on the width and slope of the roof, the truss may contain equilateral or isosceles triangles. Due to their great strength, triangles are used in the construction of bridges and bicycle frames and are visible in the Eiffel Tower.

Pentagon

The Pentagon – the headquarters of the United States Department of Defense – takes its name from its shape. The building is a pentagonal, regular pentagon.

main plate

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

fake Pentagon

A giant shopping center near Shanghai, China, is built in the shape of a regular pentagon and is sometimes referred to as a fake pentagon.

snowflakes

Each snowflake starts out as a hexagon, but temperature and humidity levels add branches and tendrils, so each snowflake looks different.

bees and wasps

Natural hexagons also include hives, where every cell in the comb that bees build to contain honey is hexagonal. Paper wasp nests also contain hexagonal cages in which they grow.

Giant sidewalk

Hexagons have also been found on the Giant’s cobbled walkway, in northeastern Ireland. It is a natural rock formation made up of about 40,000 interconnected basalt columns created as lava by a slowly cooling ancient volcanic eruption.

octagon

The octagon — the name given to the ring or cage used in Ultimate Fighting Championship (UFC) fights — takes its name from its shape. It is an octagonal regular octagon.

stop signs

Stop sign – one of the most famous road signs – another octagonal regular octagon. Although the color, text or symbols on the sign may vary, the octagonal shape of the stop sign is used in many countries around the world.