HEPTAGON

{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular heptagon
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|align=center colspan=2|
A regular heptagon
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|bgcolor=#e7dcc3|Edges and vertices||7
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|bgcolor=#e7dcc3|Schläfli symbol||{7}
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|bgcolor=#e7dcc3|Coxeter–Dynkin diagram||
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|bgcolor=#e7dcc3|Symmetry group||Dihedral (D₇)
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|bgcolor=#e7dcc3|Area
(with ''t''=edge length)||
$simeq\; 3.63391\; t^2.$
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|bgcolor=#e7dcc3|Internal angle
(degrees)||128⁵/₇°
|}
In geometry, a 'heptagon' 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 degrees. Its Schläfli symbol is {7}. The area of a regular heptagon of side length ''a'' is given by
:
The heptagon is also sometimes referred to as the ''septagon'', using "sept-" (an elision of , a Latin-derived numerical prefix, rather than , a Greek-derived numerical prefix). It is in some dictionaries.
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Contents

Construction

Heptagrams

Uses

See also

External links

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 angle
trisector. 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.

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 (2006) 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. The Brazilian 25 cents coin has a heptagon inscribed in the coin's disk.

See also

★ Heptagram

★ Heptagonal tiling

External links

★ Definition and properties of a heptagon With interactive animation

★ Approximate construction method

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