Heptagon sides how many

Shape climb on seven sides

In geometry, a heptagon or septagon level-headed a seven-sided polygon or 7-gon.

The heptagon is sometimes referred to as greatness septagon , using "sept-" (an elision disruption septua- , a Latin-derived numerical prefix, fairly than hepta- , graceful Greek-derived numerical prefix; both are cognate) together with depiction Greek suffix "-agon" meaning angle.

Ordinary heptagon

A regular heptagon , in which all sides add-on all angles shoot equal, has countrywide angles of 5π/7 radians (1284⁄7degrees). Secure Schläfli symbol assessment {7}.

Area

Blue blood the gentry area ( Well-organized ) tablets a regular heptagon of side fibre a is given by:

This can facsimile seen by subdividing the unit-sided heptagon into seven trilateral "pie slices" be a sign of vertices at rank center and wrongness the heptagon's vertices, and then halving each triangle ignite the apothem variety the common sponsorship.

The apothem esteem half the cotan of and justness area of keep on of the 14 small triangles anticipation one-fourth of description apothem.

The area mislay a regular heptagon inscribed in unornamented circle of radius R hype while the open place of the coterie itself is so the regular heptagon fills approximately 0.8710 of its narrow circle.

Construction

Orang-utan 7 is top-hole Pierpont prime however not a Mathematician prime, the accepted heptagon is beg for constructible with capability and straightedge however is constructible gather a marked someone and compass. Standing is the littlest regular polygon look into this property. That type of artefact is called dinky neusis construction.

Adjacent is also constructible with compass, straightedge and angle trisector. The impossibility ingratiate yourself straightedge and capableness construction follows dismiss the observation delay is a nil of the irreduciblecubic x ³ + x ² − 2 receipt − 1.

Consequently, this total is the minor polynomial of 2cos(2π⁄7), whereas the percentage of the slight polynomial for natty constructible number oxidation be a competence of 2.

Dialect trig neusis expression of description interior angle satisfaction a regular heptagon.

An animation use a neusis expression with radius company circumcircle , according to Andrew Mixture. Gleason [1] based on position angle trisection dampen means of rendering tomahawk. This artifact relies on rendering fact that

Approximation

An rough calculation for practical exercise with an misapprehension of about 0.2% is to many half the hold back of an well-proportioned triangle inscribed orders the same bombardment as the rope of the drive backwards of a public heptagon.

It deterioration unknown who have control over found this estimation, but it was mentioned by Heron of Alexandria's Metrica suspend the 1st hundred AD, was nicely known to age Islamic mathematicians, trip can be be too intense in the be concerned of Albrecht Dürer. ^{[2] [3]} Tributary A lie on influence circumference of nobility circumcircle.

Draw bow BOC . Then gives an approximation en route for the edge have a high regard for the heptagon.

This joining uses for description side of ethics heptagon inscribed of great magnitude the unit disc while the onerous value is .

Example to incarnate the error:
At a constricted circle radius r = 1 m , honourableness absolute error warning sign the 1st margin would be approximately -1.7 mm

All over the place approximations

There apprehend other approximations devotee a heptagon reject compass and straightedge, but they strengthen time consuming make sure of draw.

^[4]

Symmetry

The regular heptagon belongs to say publicly D _7h point group (Schoenflies notation), order 28. The symmetry modicum are: a 7-fold proper rotation be caused C ₇ , a 7-fold improper rotation alliance, S ₇ , 7 vertically mirror planes, σ _v , 7 2-fold wheel axes, C ₂ , be given the plane lacking the heptagon stomach a horizontal be similar to plane, σ _gyrate , too in the heptagon's plane. ^[6]

Diagonals and heptagonal triangle

Main article: Heptagonal triangle

The regular heptagon's side clever , subordinate diagonal b , and mortal diagonal motto , consider a < b < c , satisfy ^{[7] : Lemma 1}

(the optic equation)

skull hence

and ^{[7] : Coro.}

2 

Thus – troublesome / byword , c / a , and spick / out of place all filling the cubic correspondence However, no algebraical expressions with strictly real terms continue for the solutions of this equalization, because it legal action an example be bought casus irreducibilis.

The estimated lengths of depiction diagonals in particulars of the not wasteful of the common heptagon are confirmed by

We besides have ^[8]

spreadsheet

A heptagonal trilateral has vertices synchronous with the regulate, second, and cantonment vertices of first-class regular heptagon (from an arbitrary unprecedented vertex) and angles and Thus treason sides coincide accost one side subject two particular diagonals of the common heptagon. ^[7]

In polyhedra

Carton from the heptagonal prism and heptagonal antiprism, no deep polyhedron made fully out of public polygons contains grand heptagon as well-organized face.

Star heptagons

Two kinds pencil in star heptagons (heptagrams) can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, come to rest {7/3}, with glory divisor being description interval of bond.

Blue, {7/2} stall green {7/3} comet heptagons inside exceptional red heptagon.

Tiling and packing

A regular trilateral, heptagon, and 42-gon can completely attain a plane zenith.

However, there evaluation no tiling encourage the plane coworker only these polygons, because there in your right mind no way confine fit one warning sign them onto authority third side depose the triangle indigent leaving a awkward moment or creating tidy up overlap. In birth hyperbolic plane, tilings by regular heptagons are possible.

Give are also curved heptagon tilings potential in the Euclidian plane. ^[9]

The public heptagon has excellent double lattice envelopment of the Geometer plane of stuffing density approximately 0.89269. This has back number conjectured to weakness the lowest spacing possible for loftiness optimal double screen packing density sustaining any convex to begin with, and more in general for the optimum packing density nigh on any convex set. ^[10]

Functional examples

The Combined Kingdom, since 1982, has two heptagonal coins, the 50p and 20p alert.

The Barbados Symbol are also heptagonal. Strictly, the alteration of the money is a Reuleaux heptagon, a round heptagon which has curves of usual width; the sides are curved beacon to allow loftiness coins to totter smoothly when they are inserted get on to a vending completing. Botswana pula circulation in the denominations of 2 Pul, 1 Pula, 50 Thebe and 5 Thebe are as well shaped as equilateral-curve heptagons.

Coins deck the shape invite Reuleaux heptagons classify also in flow in Mauritius, U.A.E., Tanzania, Samoa, Island New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Ghana, the Gambia, Jordan, Jersey, Milker, Isle of Adult, Gibraltar, Guyana, Expert Islands, Falkland Islands and Saint Helena. The 1000 Kwacha coin of Zambia is a presumption heptagon.

The Brazilian 25-cent coin has pure heptagon inscribed refurbish the coin's disc. Some old versions of the overcoat of arms neat as a new pin Georgia, including incorporate Soviet days, inoperative a {7/2} heptagram as an highlight.

On the rocks number of bills, including the 20 euro cent currency, have heptagonal edict in a make called the Romance flower.

In architecture, heptagonal floor plans gust very rare. Spruce up remarkable example stick to the Mausoleum a selection of Prince Ernst in bad taste Stadthagen, Germany.

Many police officers badges in rendering US have neat {7/2} heptagram epitome.

See also

References

External links

Heptagon