Polygon

In geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure made of line segments that connect to form a closed polygonal chain. These lines are called the polygon's edges or sides. 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. This makes the polygon the boundary of a flat, enclosed area, like a solid shape. Such shapes are important in mathematics and design.

Some polygons can cross over themselves, creating interesting shapes called star polygons or other self-intersecting polygons. These special 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. This means that polygons are part of a bigger family of shapes that mathematicians study.

Etymology

The word polygon comes from ancient Greek. It is made from two words: πολύς (polús), meaning "many," and γωνία (gōnía), meaning "angle." Some people think the part "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 the same, 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. Polygons are flat shapes made of straight lines that close to form a shape.

Polygons have corners called vertices and sides called edges. Each corner has two angles: an interior angle inside the shape, and an 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

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

Other types include skew polygons, which move in a zigzag way 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." Polygons get their names from how many sides they have. We use Greek prefixes and add 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, we usually just use numbers, like 17-gon. But some special polygons have unique names, such as the pentagram. This 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 appear in nature in many cool ways. You can spot them on the flat sides of crystals, where the angles between the sides change depending on the type of mineral.

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

Computer graphics

In computer graphics, polygons are simple shapes that help create and display images. They are saved with information about their corners, colors, and textures. Surfaces are built from many connected polygons, called a polygon mesh.

When a scene is made, the computer chooses the needed polygons and shows them on the screen in the correct positions and angles. This makes pictures look three-dimensional, even though the polygons are flat.