Pentagon

A pentagon is a shape in geometry that has five sides. The word "pentagon" comes from Greek words meaning "five" and "angle." In any simple pentagon, the total amount of all its inside angles adds up to 540 degrees.

Pentagons can be simple shapes without any lines crossing, or they can be self-intersecting, where the lines cross each other. One special kind of self-intersecting pentagon is called a pentagram, which looks like a five-pointed star. Pentagon shapes appear in many areas of mathematics, art, and nature.

Regular pentagons

A regular pentagon has five equal sides and five equal angles. Each interior angle measures 108°. Regular pentagons have lines of symmetry and rotational symmetry. The diagonals of a regular pentagon relate to its sides through the golden ratio, a special number that appears in many areas of mathematics and nature.

Regular pentagons can be constructed using tools like a compass and straightedge, or even by folding paper. They have specific symmetry properties, meaning they look the same after certain rotations or reflections. A related shape is the pentagram, which is a star-shaped pentagon formed by connecting the vertices of a regular pentagon in a certain way.

Equilateral pentagons

Main article: Equilateral pentagon

An equilateral pentagon is a shape with five sides that are all the same length. Unlike a regular pentagon, which has both equal sides and equal angles, an equilateral pentagon can have different angles, creating many different kinds of five-sided shapes.

Cyclic pentagons

See also: Cyclic polygon

A cyclic pentagon is a special type of pentagon where a circle, called the circumcircle, passes through all five of its corners. The regular pentagon is an example of this kind. Some cyclic pentagons have sides and areas that are rational numbers, and these are known as Robbins pentagons.

General convex pentagons

A convex pentagon is a five-sided shape where all the interior angles are less than 180 degrees, and none of the sides curve inward. In any convex pentagon, there is a special math rule that relates the lengths of the sides and the lengths of the lines connecting opposite corners, called diagonals. This rule helps mathematicians understand more about the shape and how its parts fit together.

Pentagons in tiling

A regular pentagon cannot be used to create patterns where the shapes fit together without gaps. This is because the angles of a regular pentagon do not divide evenly into 360°, the total angle around a point.

However, there are special kinds of pentagons that can fit together to cover a flat surface completely. Scientists have found 15 different types of pentagons that can do this, though most of these do not have any symmetry.

Pentagons in polyhedra

Pentagons are important shapes in three-dimensional solids called polyhedra. Many polyhedra have pentagons as their faces. For example, a dodecahedron is a well-known solid with 12 regular pentagonal faces. These shapes help us understand the geometry of many natural and man-made objects.

Pentagons in nature

Pentagons appear in many parts of nature. In plants, you can find pentagonal shapes in flowers like morning glories and fruits such as the apple and starfruit. Some animals, like sea stars and sea urchins, also show fivefold symmetry.

Even in minerals, pentagons can be seen. For example, certain crystals like pyrite have many pentagonal faces, showing how nature uses this shape in many different ways.

Other examples

Here are two places where you can see pentagons in everyday life:

• The Pentagon, the main office building for the United States Department of Defense.
• Home plate of a baseball field, which is shaped like a pentagon.