In
geometry, a
heptagon
(or
septagon) is a
polygon
with seven sides and seven
angles. In a
regular heptagon, in which all sides
and all angles are equal, the sides meet at an angle of 5π/7
radians, 128.5714286
degree. Its
Schläfli symbol is {7}. The area A of a
regular heptagon of side length
a is given by
 A = \frac{7}{4}a^2 \cot \frac{\pi}{7} \simeq 3.63391 a^2.
The heptagon is also sometimes referred to as the
septagon, using "sept" (an elision of
septua, a
Latinderived
numerical
prefix, rather than
hepta, a
Greekderived numerical prefix). The
OED lists "septagon" as
meaning "heptagonal".
Construction
A 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 angletrisector. The
impossibility of straightedge and compass construction follows from
the observation that 2cos(2π/7) ≈ 1.247 is a zero of the
irreducible cubic x^{3} +
x^{2}  2
x  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.
A Neusis construction of the
interior angle in a regular heptagon.

A Neusis construction of the
interior angle in a regular heptagon.
(method by John Horton
Conway

Approximation
A decent approximation for practical use with an accuracy of
\approx 0.2% is shown in the drawing. Let A lie on the
circumference of the circumcircle. Draw arc BOC. Then BD = {1 \over
2}BC gives an approximation for the edge of the heptagon.
Heptagrams
Two kinds of
heptagrams 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}
heptagrams inside a red heptagon.
Uses
The
United
Kingdom currently (2009) 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
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
equilateralcurve heptagons.
The Brazilian 25 cents coin has a heptagon inscribed in the
coin's disk. Some old versions of
Coat of arms of Georgia
including
Soviet
days had used {7/2} heptagram as an element.
Regular heptagons can tile the
hyperbolic plane, as shown in this
Poincaré disk model projection:
heptagonal tiling
See also
External links