Polygon

In geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure made up of line segments connected to form a closed polygonal chain. These line segments are called the polygon's edges or sides, and the points where two edges meet are its vertices or corners. For example, a triangle is a special type of polygon with three sides, often called a 3-gon.

A simple polygon does not cross over itself. The only points where its edges meet are at the corners where two edges end and begin again. This makes the polygon the boundary of a flat, enclosed area, which can be thought of as a solid shape. Such shapes are important in many areas of mathematics and design.

Some polygons can cross over themselves, creating interesting shapes called star polygons or other self-intersecting polygons. These special kinds of polygons are studied in geometry and can be found in art, architecture, and nature.

Polygons are two-dimensional examples of a more general idea called a polytope, which can exist in any number of dimensions. This means that polygons are part of a bigger family of shapes that mathematicians study.

Etymology

The word polygon comes from ancient Greek. It combines two words: πολύς (polús), meaning "much" or "many," and γωνία (gōnía), meaning "corner" or "angle." Some people think the part of the word that means "gon" might also come from another Greek word, γόνυ (gónu), which means "knee."

Classification

Polygons can be grouped by how many sides they have and by their shape.

Polygons are often described by whether all their sides and angles are equal, and whether their sides curve inward or outward. Some polygons are simple, meaning their sides do not cross each other, while others are self-intersecting, where the sides cross over themselves. Regular polygons have both equal sides and equal angles, making them very symmetrical.

Properties and formulas

Euclidean geometry helps us understand polygons, which are flat shapes made of straight lines that close to form a shape.

Polygons have corners called vertices and sides called edges. The two most important angles at each corner are the interior angle, inside the shape, and the exterior angle, outside the shape. For any polygon with n sides, the sum of the interior angles is (n − 2) × 180 degrees. The sum of the exterior angles is always 360 degrees.

We can find the area of a polygon using special formulas. For simple polygons that don’t cross themselves, one common method is the shoelace formula. For regular polygons, which have all sides and angles equal, there are simpler formulas that use the radius of the circle surrounding the polygon.

Generalizations

The idea of a polygon can mean different things in special situations. For example, a spherical polygon is made from arcs on the surface of a ball and can have shapes that are impossible on a flat surface. These are useful for making maps and building special solid shapes called uniform polyhedra.

Other types include skew polygons, which zigzag in three dimensions, and shapes that go on forever without ending. There are also polygons with holes inside them and complex polygons that exist in imaginary spaces.

Naming

The word polygon comes from ancient Greek, meaning "many-angled." Individual polygons are named based on the number of sides they have, using Greek prefixes with the ending ‑gon. For example, a shape with five sides is called a pentagon, and one with twelve sides is a dodecagon.

For polygons with more than twelve sides, mathematicians usually just use numbers, like 17-gon. However, some special polygons have unique names, such as the pentagram, which is a type of star-shaped pentagon.

History

Polygons have been known since ancient times. The regular polygons were studied by the ancient Greeks. A special shape called the pentagram, which is a star-shaped polygon, was found on a piece of pottery from around the 7th century B.C.

Later, mathematicians continued to explore polygons. In the 1300s, a person named Thomas Bradwardine began studying more complex polygon shapes. In 1952, a mathematician named Geoffrey Colin Shephard expanded the idea of polygons to include imaginary numbers, creating something called complex polygons.

In nature

Polygons can be found in nature in many interesting places. You can see them in the flat sides of crystals, where the angles between the sides depend on the type of mineral.

Regular hexagons appear when lava cools and forms tightly packed columns of basalt. Famous examples include the Giant's Causeway in Northern Ireland and the Devil's Postpile in California. In biology, the wax honeycomb made by bees is made up of hexagons, and each cell’s sides and base are also polygons.

Computer graphics

In computer graphics, polygons are basic shapes used to build and show images. They are stored with details about their corners, colors, and textures. Surfaces are made from many connected polygons, called a polygon mesh.

When a scene is created, the computer picks the needed polygons and shows them on the screen in the right positions and angles. This helps make pictures look three-dimensional even though the polygons themselves are flat.