In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a heptagon or septagon is a seven-sided
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
or 7-gon.
The heptagon is sometimes referred to as the septagon, using ''
septa-'' (an
elision
In linguistics, an elision or deletion is the omission of one or more sounds (such as a vowel, a consonant, or a whole syllable) in a word or phrase. However, these terms are also used to refer more narrowly to cases where two words are run to ...
of ''
septua-''), a
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
-derived
numerical prefix, rather than ''
hepta-'', a
Greek
Greek may refer to:
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group
*Greek language, a branch of the Indo-European language family
**Proto-Greek language, the assumed last common ancestor of all kno ...
-derived numerical prefix (both are cognate), together with the suffix ''-gon'' for , meaning angle.
Regular heptagon
A
regular heptagon, in which all sides and all angles are equal, has
internal angle
In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
s of 5π/7
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s (128
degrees). Its
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
is .
Area
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
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...
as the common side. The apothem is half the
cotangent
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
of
and the area of each of the 14 small triangles is one-fourth of the apothem.
The area of a regular heptagon
inscribed
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same th ...
in a circle of
radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
''R'' is
while the area of the circle itself is
thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.
Construction
As 7 is a
Pierpont prime
In number theory, a Pierpont prime is a prime number of the form
2^u\cdot 3^v + 1\,
for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...
but not a
Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...
, the regular heptagon is not
constructible with
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
but is constructible with a marked
ruler
A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...
and compass. It is the smallest regular polygon with this property. This type of construction is called a
neusis construction. It is also constructible with compass, straightedge and
angle trisector. The impossibility of straightedge and compass construction follows from the observation that
is a zero of the
irreducible cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
. Consequently, this polynomial is the
minimal polynomial of whereas the degree of the minimal polynomial for a
constructible number must be a power of 2.
Approximation
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
Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
. Let ''A'' lie on the circumference of the circumcircle. Draw arc ''BOC''. Then
gives an approximation for the edge of the heptagon.
This approximation uses
for the side of the heptagon inscribed in the unit circle while the exact value is
.
''Example to illustrate the error:
At a circumscribed circle radius ''r = 1 m'', the absolute error of the 1st side would be ''approximately -1.7 mm''
Other approximations
There are other approximations of a heptagon using compass and straightedge, but they are time consuming to draw.
Symmetry

The ''regular heptagon'' belongs to the
D7h point group
In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...
(
Schoenflies notation), order 28. The symmetry elements are: a 7-fold proper rotation axis C
7, a 7-fold improper rotation axis, S
7, 7 vertical mirror planes, σ
v, 7 2-fold rotation axes, C
2, in the plane of the heptagon and a horizontal mirror plane, σ
h, also in the heptagon's plane.
Diagonals and heptagonal triangle

The regular heptagon's side ''a'', shorter
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
''b'', and longer diagonal ''c'', with ''a''<''b''<''c'', satisfy
[Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", '' Forum Geometricorum'' 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf]
:
:
:
:
(the
optic equation
In number theory, the optic equation is an equation that requires the sum of the multiplicative inverse, reciprocals of two positive integers and to equal the reciprocal of a third positive integer :Dickson, L. E., ''History of the Theory of N ...
)
and hence
:
and
[
:
:
:
Thus –''b''/''c'', ''c''/''a'', and ''a''/''b'' all satisfy the ]cubic equation
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
However, no algebraic expression
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations:
addition (+), subtraction (-), multiplication (×), division (÷), whole number pow ...
s 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
:
:
:
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 and Thus its sides coincide with one side and two particular diagonals of the regular heptagon.[
]
In polyhedra
Apart from the heptagonal prism and heptagonal antiprism, no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.
Star heptagons
Two kinds of star heptagons (heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
s) can be constructed from regular heptagons, labeled by Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
s , and , with the divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
being the interval of connection.

Blue, and green star heptagons inside a red heptagon.
Tiling and packing
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.
The regular heptagon has a double lattice packing of the Euclidean plane of packing density approximately 0.89269. This has been conjectured to be the lowest density possible for the optimal double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set.
Empirical examples
The United Kingdom, since 1982, has two heptagonal coin
A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by ...
s, the 50p and 20p pieces. The Barbados Dollar are also heptagonal. Strictly, the shape of the coins is a Reuleaux heptagon, a curvilinear
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inv ...
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
A vending machine is an automated machine that dispenses items such as snacks, beverages, cigarettes, and lottery tickets to consumers after cash, a credit card, or other forms of payment are inserted into the machine or payment is otherwise m ...
. Botswana pula
The Pula (also known as the Botswana dollar) is the currency of Botswana. It has the ISO 4217 code ''BWP'' and is subdivided into 100 ''thebe''. ''Pula'' literally means "rain" in Setswana, because rain is very scarce in Botswana—home to much o ...
coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. 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.
The Brazil
Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
ian 25-cent coin has a heptagon inscribed in the coin's disk. Some old versions of the coat of arms of Georgia, including in Soviet days, used a heptagram as an element.
A number of coins, including the 20 euro cent coin, have heptagonal symmetry in a shape called the Spanish flower.
In architecture, heptagonal floor plans are very rare. A remarkable example is the Mausoleum of Prince Ernst in Stadthagen, Germany
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
.
Many police badges in the US have a heptagram outline.
See also
* Heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
*Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
References
External links
Definition and properties of a heptagon
With interactive animation
Heptagon according Johnson
Another approximate construction method
Recently discovered and highly accurate approximation for the construction of a regular heptagon.
* Heptagon, an approximating construction as an animation
* A heptagon with a given side, an approximating construction as an animation
{{Polygons
Polygons by the number of sides
7 (number)
Elementary shapes
Heptagon