Heptagon

In geometry, a heptagon is a seven-sided polygon or 7-gon. It is a basic shape studied in math and appears in many real-world designs and patterns.

The heptagon is sometimes called a septagon, using septa- (a shortened form of septua-), which comes from Latin. This prefix is used instead of hepta-, which has Greek origins. Both prefixes mean the number seven, and they are related in language. The word ends with ‑gon, a suffix from the Greek word γωνἰα, romanized as gonía, meaning "angle."

Heptagons can be found in architecture, art, and nature. Their unique shape makes them interesting for both learning and creative projects.

Regular heptagon

A regular heptagon is a seven-sided polygon where all sides and angles are equal. Each internal angle measures 128 4⁄7 degrees or about 128.57°. The shape belongs to a specific symmetry group and has unique properties regarding its diagonals and construction.

The area of a regular heptagon can be calculated if the length of one side is known. For a heptagon inscribed in a circle, the shape fills roughly 87.10% of the circle's area. While it cannot be constructed using only a compass and straightedge, special tools or methods allow its creation. The heptagon also relates to special triangles called heptagonal triangles, which use its sides and diagonals in their structure.

A neusis construction of the interior angle in a regular heptagon.

An animation from a neusis construction with radius of circumcircle O A ¯ = 6 {\displaystyle {\overline {OA}}=6} , according to Andrew M. Gleason based on the angle trisection by means of the tomahawk. This construction relies on the fact that cos ⁡ ( 2 π 7 ) = 1 6 ( 2 7 cos ⁡ ( 1 3 arctan ⁡ 3 3   ) − 1 ) . {\displaystyle \cos \left({\tfrac {2\pi }{7}}\right)={\tfrac {1}{6}}\left(2{\sqrt {7}}\cos \left({\tfrac {1}{3}}\arctan 3{\sqrt {3}}~\right)-1\right).}

Star heptagons

Two special star-shaped heptagons can be made from regular heptagons. These are called {7/2} and {7/3}, based on a special math rule for connecting the points. You can see them as blue and green stars inside a red heptagon shape.

These star heptagons are interesting because they show how shapes can be rearranged in new and beautiful ways.

Tiling and packing

A regular triangle, heptagon, and 42-gon can fit together at a point without leaving a gap. However, they cannot tile the entire plane by themselves because fitting them together leaves spaces. In the hyperbolic plane, tilings using regular heptagons are possible. There are also ways to tile the plane using concave heptagons.

The regular heptagon can be packed closely together in a pattern that covers about 89.269% of the plane, which is thought to be one of the least dense packings possible for shapes of this type.

Empirical examples

Some coins from Zambia are shaped like heptagons, which have seven sides. Many countries use coins shaped like heptagons to help them roll smoothly in machines. These include coins from the United Kingdom, Barbados, and Botswana, among others.

Buildings like the Mausoleum of Prince Ernst in Stadthagen, Germany, also have heptagonal shapes. Some old designs of the coat of arms of Georgia included a special seven-pointed star shape.