Polygon Geometry: Pentagons, Hexagons, and Dodecagons (2024)

A special kind of polygon can have angles that aren't all equal. In this case, it's called an irregular polygon.

About Polygons

The name polygon comes from two Greek words:

• Poly, which meansmany
• Gon,which meansangle

Shapes That ArePolygons

• Trigon (triangle): 3 sides
• Tetragon (square): 4 sides
• Pentagons: 5 sides
• Hexagon: 6 sides
• Heptagon: 7 sides
• Octagons: 8 sides
• Nonagon: 9 sides
• Decagon: 10 sides
• Undecagon: 11 sides
• Dodecagons: 12 sides

How Polygons Are Named

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

The common name for most polygons is theGreek prefixfor "sides"attached to the Greek word for corner (gon).

Examples of this 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 namingscheme. Most notably with words more commonly used for some polygons:

• Triangle:Uses the Greek prefixTri,but instead of theGreekgon,the Latinangleis used. Trigon is the correct geometrical name but is rarely used.
• Quadrilateral:Derivedfromthe Latin prefixquadri,meaning four, attached to the word lateral,which is another Latin word meaningside.
• Square: Sometimes, a four-sided polygon (a square) is referred to as aquadrangleortetragon.

N-Gons

Polygons with more than 10sides are encountered infrequently but follow the same Greek naming convention. So,a 100-sided polygon isreferred to as a hectogon.

However, in mathematics, pentagons are sometimes more conveniently referred to as n-gons:

• 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 a polygon can have.

Asthe size of the interior angles of a polygon gets larger, and the length of its sides gets shorter, thepolygon approaches a circle, but it never quite gets there.

Classifying Polygons

Regular vs. Irregular Polygons

Polygons are classified based on whether or not all angles or sides are equal.

• Regular polygon: All of the angles are of equal size, and all the sides are equal in length.
• Irregular polygon: Does not have equal-sized angles or sides of equal length.

Convex vs.Concave Polygons

A secondway to classify polygons is by the size of their internal angles.

• Convex polygons: Haveno internal anglesgreaterthan 180°.
• Concave polygons: Have at least one internal angle that isgreater than 180°.

Simple vs. Complex Polygons

Another way to classify polygons is the way the lines forming the polygon intersect.

• Simple polygons: The lines connect or intersect only once — at the vertices.
• Complex polygons: The lines intersect more than once.

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

For example:

• A regular-shapedhexagonis a six-sided, simple polygon.
• A star-shaped hexagramis a six-sided, complex polygoncreated by overlapping twoequilateraltriangles.

Sum of the Interior Angles Rule

As a rule, each timea 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 hexagoncan be foundby using the formula:

(6 - 2)× 180° =720°

How Many Triangles in That Polygon?

The above interior angleformula is derived by dividing apolygon into triangles, and this number can be found with the calculation:

n - 2

In this formula, n is equal to the number of sides of the polygon.

A hexagon (sixsides)can be divided into fourtriangles (6 - 2) and a dodecagon into 10 triangles (12 - 2).

Angle Size forRegularPolygons

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

For a regular six-sidedhexagon, each angle is:

720° ÷ 6 = 120°

Some Well-Known Polygons

Well-know polygons include:

Trusses

Roof trussesare often triangular. Depending on the width andpitch of the roof, the truss might incorporateequilateral orisosceles triangles. Because of their greatstrength, triangles areused in the construction of bridges and bicycleframes. 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. The building is a five-sided, regular pentagon.

Home Plate

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

The Fake Pentagon

A giant shopping mall near Shanghai,China,isbuilt in the shape of a regular pentagonandis sometimes calledthe Fake Pentagon.

Snowflakes

Every snowflake starts out as a hexagon, buttemperature and moisture levels addbranches and tendrils so that eachone ends up looking different.

Bees and Wasps

Natural hexagons alsoinclude beehives, where eachcellin a honeycomb that the bees construct to hold honey is hexagonal. The nests of paper wasps also containhexagonal cells where they raise their young.

The Giant's Causeway

Hexagonsare also found intheGiant's Causewaylocated in northeastIreland. It is a natural rock formation composed of about 40,000 interlockingbasalt columnsthatwere 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 regularoctagon.

Stop Signs

The stop sign — one of the most familiar traffic signs — is another eight-sided regular octagon. Although the color, wording, or symbols on the sign may vary, the octagonal shape for the stop sign is used in many countries around the world.

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