The heptagon is sometimes referred to as theseptagon, usingsepta- (anelision ofseptua-), aLatin-derivednumerical prefix, rather thanhepta-, aGreek-derived numerical prefix (both are cognate), together with the suffix-gon forGreek:γωνἰα,romanized: gonía, meaning angle.
The area (A) of a regular heptagon of side lengtha is given by:
This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" withvertices at the center and at the heptagon's vertices, and then halving each triangle using theapothem as the common side. The apothem is half thecotangent of, and the area of each of the 14 small triangles is one-fourth of the apothem.
The area of a regular heptagoninscribed in a circle ofradiusR is while the area of the circle itself is thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.
As 7 is aPierpont prime but not aFermat prime, the regular heptagon is notconstructible withcompass and straightedge but is constructible with a markedruler and compass. It is the smallest regular polygon with this property. This type of construction is called aneusis construction. It is also constructible with compass, straightedge andangle trisector. The impossibility of straightedge and compass construction follows from the observation that is a zero of theirreduciblecubicx3 +x2 − 2x − 1. Consequently, this polynomial is theminimal polynomial of, whereas the degree of the minimal polynomial for aconstructible number must be a power of 2.
Aneusis construction of the interior angle in a regular heptagon.
An animation from a neusis construction with radius of circumcircle, according toAndrew M. Gleason[1] based on theangle trisection by means of thetomahawk. This construction relies on the fact that
Heptagon withgiven side length: An animation from aneusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson).
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 byHeron of Alexandria'sMetrica in the 1st century AD, was well known to medieval Islamic mathematicians, and can be found in the work ofAlbrecht Dürer.[2][3] LetA lie on the circumference of the circumcircle. Draw arcBOC. 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 radiusr = 1 m, the absolute error of the first side would be approximately−1.7 mm
Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.[5]
Theregular heptagon belongs to theD7hpoint group (Schoenflies notation), order 28. The symmetry elements are: a 7-fold proper rotation axis C7, a 7-fold improper rotation axis, S7, 7 vertical mirror planes, σv, 7 2-fold rotation axes, C2, in the plane of the heptagon and a horizontal mirror plane, σh, also in the heptagon's plane.[6]
Aheptagonal triangle hasvertices 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 particulardiagonals of the regular heptagon.[7]
Two kinds of star heptagons (heptagrams) can be constructed from regular heptagons, labeled bySchläfli symbols {7/2}, and {7/3}, with thedivisor being the interval of connection.
Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.
A regular triangle, heptagon, and 42-gon can completelyfill 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 thehyperbolic plane, tilings by regular heptagons are possible. There are also concave heptagon tilings possible in the Euclidean plane.[9]
The densestdouble lattice packing of the Euclidean plane by regular heptagons, conjectured to have the lowest maximum packing density of any convex set
The regular heptagon has adouble 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.[10]
TheBrazilian 25-cent coin has a heptagon inscribed in the coin's disk. Some old versions of thecoat of arms of Georgia, including inSoviet days, used a {7/2} heptagram as an element.
^John H. Conway, Heidi Burgiel,Chaim Goodman-Strauss, (2008) The Symmetries of Things,ISBN978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
