Names of Polygons are decided based on the number of sides it contains. A polygon is a closed plane figure made up of segments (also called sides). Polygons are named according to the number of segments from which they are comprised

The word “geometry” comes from the Greek geos which means “earth” and metron which means “measurement”. In this chapter we will learn the differences between polygons and polyhedrons and how to calculate various lengths (perimeters), areas, and volumes (in the case of the polyhedrons) of various geometric figures.

## Polygons: An Introduction

A polygon is a closed plane figure made up of segments (also called sides). Polygons are named according to the number of segments from which they are comprised. The term n-gon is used to denote a polygon with n sides. The n is substituted, according to the number of sides that

comprise the figure, by the Greek cardinal prefixes: penta (or 5), hexa (or 6), hepta (or 7), etc. The only two exceptions are the three- and four-sided polygons which are the triangles (never “trigons”) and quadrilaterals (never “tetragons”). The number of sides is equal to the number of

angles in all polygons.

Polygons can be classified in two different ways: internal angles and lengths of sides. Depending on the measure of the internal angles, polygons can be convex, with all internal angles less than 180° (see page 211), or concave (with at least one or more internal angles greater than 180°). You can distinguish a convex polygon from a concave polygon by noting that convex polygons have all their angles pointing out, while concave polygons have at least one angle that points inward. It is impossible to construct a 3-sided (triangular) concave polygon.

## Examples of Concave Polygons

## Examples of Convex Polygons

Convex polygons can also be classified as regular (when they are equilateral―that is, all sides have the same length; and equiangular―that is, all angles have the same measure) and irregular (when at least one angle or side differs from the rest).

## Names Of Polygons: Examples of Regular Polygons:

## Examples of Irregular Polygons

The parts that make up a polygon are the following:

• **Sides**: the segments or straight lines that form the polygon.

• **Vertices**: the point where the sides meet. The sides always form an angle at each vertex.

• **Diagonals**: are the lines that join two non-consecutive angles.

Consider the following convex hexagon:

A “hexagon” is a 6-sided polygon having 6 vertices and 9 different diagonals. In general, the number of diagonals can be computed given the number of vertices, n, using this formula:

no. of diagonals = n(n – 3)/2

Diagonals that connect the vertices of a concave polygon, will sometimes lie partially or completely outside the boundaries of the object.

The polygons are named according to the number of sides they contain as given in the table below. In actual practice, instead of, for example, identifying a 40-sided polygon as a tetracontagon, most mathematicians will simply call it a 40-gon.

## Names Of Polygons And Sides:

Name of Polygon | No. Of Sides | Name of Polygon | No. Of Sides |

triangle | 3 | enneadecagon | 19 |

quadrilateral | 4 | icosagon | 20 |

pentagon | 5 | triacontagon | 30 |

hexagon | 6 | tetracontagon | 40 |

heptagon | 7 | pentacontagon | 50 |

octagon | 8 | hexacontagon | 60 |

nonagon | 9 | heptacontagon | 70 |

decagon | 10 | octacontagon | 80 |

hendecagon | 11 | enneacontagon | 90 |

dodecagon | 12 | hectagon | 100 |

triskaidecagon | 13 | chiliagon | 1000 |

tetrakaidecagon | 14 | myriagon | 10,000 |

pentadecagon | 15 | Megagon | 1,000,000 |

hexakaidecagon | 16 | googolgon | 10^100 |

heptadecagon | 17 | n-gon | n |

octakaidecagon | 18 |

## Optional Formulas:

The sum of the interior angles of a general n-sided polygon is given by the following:

Sum of interior angles = (n – 2)180°

Given an interior angle, a, the exterior angle is 180 – a and the number of sides of a regular ngon having that interior angle is given by

n=360/180−a.

Knowing n, we can find a = 180 -(360/n).

The sum of the exterior angles of a polygon is always 360 degree.

## Polyhedrons: An Introduction

All polygons by definition are two-dimensional (i.e., flat) objects. Let´s consider now threedimensional figures (i.e., solid figures that have height, width, and depth).

A polyhedron is a geometric solid in three dimensions with flat faces and straight edges. A polyhedron is a three-dimensional figure assembled with polygonal-shaped surfaces. In a manner similar to that of polygons. Polyhedrons are comprised of

• **Faces**: Plane polygon-shaped surfaces.

• **Edges**: Intersections between faces.

• **Vertices**: The vertices of the faces are the vertices of the polyhedron also. Three or more faces can meet at a vertex.

Interestingly, the number of vertices (V), edges (E), and faces (F) of a polyhedron are related according to this formula:

**V – E + F = 2**

Here are some examples of polyhedrons (or polyhedra):

## Names Of Polygons : Tetrahedron:

## Cube Or Hexahedron:

## Octahedron:

## Icosahedron:

Just as polygons are named based on the number of sides that comprise them, polyhedrons are named after the number of faces, that comprise them. The n-hedron names are generated by substituting the n with the Greek cardinal prefixes (tri—or 3, tetra—or 4, octa—or 8, dodeca—or 12, icosa—or 20) that match the number of faces that comprise the figure.

Name of polyhedron | Number of faces | Name of polyhedron | Number of faces |

trihedron | 3 | dodecahedron | 12 |

tetrahedron | 4 | tetradecahedron | 14 |

pentahedron | 5 | icosahedron | 20 |

hexahedron | 6 | icositetrahedron | 24 |

heptahedron | 7 | triacontrahedron | 30 |

octahedron | 8 | icosidodecahedron | 32 |

nonahedron | 9 | hexecontahedron | 60 |

decahedron | 10 | enneacontahedron | 90 |

undecahedron | 11 | n-hedron | n |

## Names Of Polygons : Prisms and pyramids:

There are other types of non-polyhedral solids that we are going to consider in this chapter, known as prisms and pyramids. The main difference between these and polyhedrons consists in the different shaped-faces that constitute them.

**Prisms** are three-dimensional figures with two similar (or congruent—same shape and size, but in different positions) polygon bases that rest in parallel planes. Pyramids, on the other hand, have only one polygon base in a plane with the remainder of the faces connected to a point outside this plane.

**Regular** prisms and pyramids are named after the shape of the base they have. In the figures above, for example, since the base is a square, a regular square prism and a regular square pyramid are shown.