In geometry, a heptagon is a 7-sided polygon or 7-gon.
The heptagon is also occasionally referred to as the septagon, using
"sept-" together with the Greek suffix "-agon" meaning angle.
Regular heptagon A regular heptagon, in which all sides
and all angles are equal, has internal angles of 5π/7 radians. Its Schläfli
symbol is {7}. = Area =
The area 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 apothem is half the cotangent of and the area of each of the 14 small
triangles is one-fourth of the apothem. The exact algebraic expression, starting
from the polynomial x3 + x2 − 2x − 1 is given by:
where is the imaginary unit. The area of a regular heptagon inscribed
in a circle of radius 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 but not a
Fermat prime, the regular heptagon is not constructible with compass and
straightedge but is constructible with a marked ruler and compass. 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 x3 + x2 − 2x − 1. Consequently this polynomial is
the minimal polynomial of 2cos(2π⁄7), 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 shown in the drawing. It is attributed to Albrecht
Dürer. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then
gives an approximation for the edge of the heptagon.
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
= A more accurate approximation = A regular heptagon with sides can be
inscribed in a circle of the radius with an error of less than 0.00013%.
This follows from a rational approximation of .
For the construction of S and R see: The example the division 3÷2 on the real
number line and Intercept theorem, Formulation 4th
point Example to illustrate the error:
At a circumscribed circle radius R = 1 km, S = 0.867768595041322 km
and S - Sshould = 0.867768595041322 km - 0,867767478235116 km
the absolute error of the 1st side would be approximately 1.1 mm.
= Symmetry = The regular heptagon has Dih7 symmetry,
order 14. Since 7 is a prime number there is one subgroup with dihedral
symmetry: Dih1, and 2 cyclic group symmetries: Z7, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the heptagon.
John Conway labels these by a letter and group order. Full symmetry of the
regular form is r14 and no symmetry is labeled a1. The dihedral symmetries are
divided depending on whether they pass through vertices or edges, and i when
reflection lines path through both edges and vertices. Cyclic symmetries in the
middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular
forms. Only the g7 subgroup has no degrees of freedom but can seen as
directed edges. Star heptagons
Two kinds of star heptagons can be constructed from regular heptagons,
labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the
interval of connection. Blue, {7/2} and green {7/3} star
heptagons inside a red heptagon. Many police badges in the US have a 7:2
heptagram outline. Uses
The United Kingdom currently has two heptagonal coins, the 50p and 20p
pieces, and the Barbados Dollar is also heptagonal. The 20-eurocent coin has
cavities placed similarly. Strictly, the shape of the coins is a Reuleaux
heptagon, a curvilinear heptagon to make them curves of constant width: the sides
are curved outwards so that the coin will roll smoothly in vending machines.
Botswana pula 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 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 Brazilian 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 {7/2} heptagram as an element.
In architecture, heptagonal floor plans are very rare. A remarkable example is
the Mausoleum of Prince Ernst in Stadthagen, Germany.
Apart from the heptagonal prism and heptagonal antiprism, no convex
polyhedron made entirely out of regular polygons contains a heptagon as a face.
Regular heptagons can tile the hyperbolic plane, as shown in this
Poincaré disk model projection: heptagonal tiling
Graphs The K7 complete graph is often drawn as
a regular heptagon with all 21 edges connected. This graph also represents an
orthographic projection of the 7 vertices and 21 edges of the 6-simplex.
The regular skew polygon around the perimeter is called the petrie polygon.
Heptagon in natural structures See also
Heptagram References
External links Definition and properties of a heptagon
With interactive animation Weisstein, Eric W., "Heptagon",
MathWorld. Another approximate construction method
Polygons – Heptagons Recently discovered and highly accurate
approximation for the construction of a regular heptagon.