Puzzling Polygon Facts Every Aspiring Mathematician Will Adore

Any two-dimensional closed plane figure with sides and not curves is a polygon.

The term polygon originated from the Greek language, where 'poly' means many, and 'gonia' means angle. Triangles, quadrilaterals, pentagons, and octagons are all polygons.

Studying geometry as part of mathematics is very interesting and amusing. When straight-line segments connect to each other to form a closed plane figure, it is called a polygon. In Euclidean geometry, which is also called flat geometry, the smallest possible polygon has three sides and is called a triangle.

Types Of Polygon

Polygons can be regular or irregular polygons, convex or concave polygons, or simple or complex polygons.

Regular polygons have all equal sides and angles. If the sides are unequal in length, they are irregular polygons. An equilateral triangle or a square with four sides are regular polygons, whereas a solid arrow on a signboard is an example of an irregular polygon.

If all angles inside a polygon are less than 180 degrees, it is called a convex polygon. Squares and rectangles are examples of a convex polygon.

If any one of the interior angles is greater than 180 degrees, it is called a concave polygon. A rhombus is an example of a concave polygon. Concave polygons are very common and have a more irregular shape, and a concave polygon is also called a non-convex polygon.

Any polygon that does not intersect itself is a simple polygon. If any of the edges intersect themselves, it is a complex polygon.

A star drawn with only external sides is a simple polygon, and if it is drawn with all its sides inside, they intersect each other and become a complex polygon. Complex polygons often have an irregular shape.

Properties Of Polygon

Any polygon study requires understanding the following three key properties: the number of sides of polygons, angles between the sides or edges, and length of the sides or edges.

A polygon is defined by the number of sides it has. Triangle is the smallest polygon with three sides. Equal-sided triangles are called equilateral triangles.

If two sides are equal, they are isosceles triangles, and all three sides being different implies that they are scalene triangles. A four-sided polygon is a quadrilateral. Squares and rectangles are all examples of this polygon.

Square is a regular polygon because of its equal sides. Five sides make the polygon a pentagon, six sides make it a hexagon, seven sides make it a heptagon, and so on. A thousand-sided polygon is called a chiliagon.

In their discussions, philosophers like Immanuel Kant, David Hume, and Descartes, referred to a chiliagon. A million-sided polygon is called a megagon and describes a philosophical concept that cannot be visualized. It is also considered to explain the convergence of several regular polygons as a circle.

The angles between the sides of polygons also constitute interesting polygon facts. For any polygon, the sum of all internal angles can be calculated with a formula:

The sum of internal angles = 180 degrees x (number of sides - 2)

Along with the number of sides and angles, the length of each side is also important. For a regular polygon, measuring one side is sufficient.

Polygons In Computer Graphics

Polygons have a vital role in computer graphics. In modeling, imaging, and rendering, polygons are used as basic entities. All attributes of polygons are defined in the form of arrays.

Vertices, sides, length, color, angles, and texture are all defined as arrays in the database. The images are stored in the form of a polygon mesh as a tessellation.

A tessellation is a recurring symmetrical, interlocking shape pattern and is often complex. These structures of polygon images are called from the database to active memory and then to display screen to be viewed as rendered scenes. These two-dimensional polygons are oriented so that they are viewed as three-dimensional visual scenes.

In computer graphics, an important requirement is to determine if a given point is inside or outside a polygon. A test called point in polygon test or inside test is conducted. Polygon filling is another important requirement where the polygon is filled with color. Several algorithms such as Boundary fill, Flood fill, or Scalene fills are used.

Angles In Polygon

Every polygon has two types of angles: interior angle and exterior angle. Angles formed by the lines or edges of the polygon on the inside are called interior angles.

It is measured at the vertex, on the inside of the polygon. Angles for outside of the polygon when one of the edges is extended are called exterior angles. Some angle properties of regular polygons are:

Sum total of all exterior angles is 360 degrees.

If a polygon has n number of sides, each exterior angle is 360 degrees/n.

Sum total of all interior angles is (n-2) x 180 degrees for a regular polygon with n being the number of sides.

Each interior angle is calculated as (n-2) x 180 degrees/n.

FAQs

Q: What is special about a regular polygon?

A: A regular polygon has all sides and angles equal.

Q: How many sides are on a polygon?

A: A polygon has a minimum of three sides and infinite maximum sides.

Q: What are the 20 polygons?

A: Triangle (three sides), quadrilateral (four sides), pentagon (five sides), hexagon (six sides), heptagon (seven sides), octagon (eight sides), nonagon (nine sides), decagon (10 sides), hendecagon (11 sides), dodecagon (12 sides), tridecagon (13 sides), tetradecagon (14 sides), pentadecagon (15 sides), hexadecagon (16 sides), heptadecagon (17 sides), octadecagon (18 sides), enneadecagon (19 sides), icosagon (20 sides), chilliagon (one thousand sides), and megagon (one million sides).

Q; What is the polygon shape?

A: A polygon can be of any shape, which is a plane figure closed with lines and not curves.

Q: Are all polygons quadrilaterals?

A: No, only polygons with four sides are quadrilaterals.

Q: What do polygons have in common?

A: Regular polygons have equal sides and angles, which are common.