In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.

The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.

The area (A) of a regular heptagon of side length a is given by: This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side.

The area of a regular heptagon inscribed in a circle of radius R is

thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.

As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass.

It is also constructible with compass, straightedge and angle trisector.

An approximation for practical use with an error of about 0.2% is to use half the side of an equilateral triangle inscribed in the same circle as the length of the side of a regular heptagon.

It is unknown who first found this approximation, but it was mentioned by Heron of Alexandria's Metrica in the 1st century AD, was well known to medieval Islamic mathematicians, and can be found in the work of Albrecht Dürer.

for the side of the heptagon inscribed in the unit circle while the exact value is

There are other approximations of a heptagon using compass and straightedge, but they are time consuming to draw.

[4] The regular heptagon belongs to the D7h point group (Schoenflies notation), order 28.

However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis.

The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by We also have[8] and A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles

Two kinds of star heptagons (heptagrams) can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection.

A regular triangle, heptagon, and 42-gon can completely fill a plane vertex.

However, there is no tiling of the plane with only these polygons, because there is no way to fit one of them onto the third side of the triangle without leaving a gap or creating an overlap.

In the hyperbolic plane, tilings by regular heptagons are possible.

There are also concave heptagon tilings possible in the Euclidean plane.

[10] The United Kingdom, since 1982, has two heptagonal coins, the 50p and 20p pieces.

Strictly, the shape of the coins is a Reuleaux heptagon, a curvilinear heptagon which has curves of constant width; the sides are curved outwards to allow the coins to roll smoothly when they are inserted into a vending machine.

Coins in the shape of Reuleaux heptagons are also in circulation in Mauritius, U.A.E., Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Ghana, the Gambia, Jordan, Jersey, Guernsey, Isle of Man, Gibraltar, Guyana, Solomon Islands, Falkland Islands and Saint Helena.

The 1000 Kwacha coin of Zambia is a true heptagon.

Some old versions of the coat of arms of Georgia, including in Soviet days, used a {7/2} heptagram as an element.

A remarkable example is the Mausoleum of Prince Ernst in Stadthagen, Germany.