Pentagon

A pentagon is a geometric figure. It belongs to the group of polygons (polygons ) and is defined by five points. Called a regular pentagon, and Pentagon (from the Greek ), has five equal sides, the corners are all on a common radius.

• 3.1 Nature
• 3.2 architecture and fortifications

Mathematical relationships

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Angle

The sum of the interior angles of a regular pentagon is 540 °, and results from a general formula for polygons in which the variable is the number of vertices of the polygon has to be used ( in this case):

The angle which the two adjacent side edges in the plane, a regular pentagon with each other is (again, according to a general formula for regular polygons, see # polygon interior angles )

Surface

A pentagonal plane has a clearly identifiable area, which can be calculated by decomposing into triangles always. The surface A of the side length of a regular pentagon is five times the area of ​​a plane defined by its center and two of its corners a triangle.

General to the perimeter radius ru

Or

Side length

Or also:

For conversion, see the section on specifiable as square sine and cosine values ​​.

The golden section in the pentagon

Regular pentagon and pentagram form a basic figure, in which the ratio of the Golden Section occurs repeatedly. The side of the pentagon is located in the golden ratio to its diagonal. The diagonals with each in turn divided in the golden ratio, that is, AD is related to BD as BD to CD. The proof uses the similarity suitably selected triangles.

Construction with ruler and compass

For the regular pentagon, there is a mathematically exact construction to determine the side length. The following are the notes to the adjacent figure (click image shows enlargement and the colors used for better illustration ):

Calculation of Construction:

Corresponding exactly to the factor in the above formula for the page length.

The side edges of the triangle MEF correspond exactly to the side lengths of the regular hexagon (ME), the regular pentagon (EF ) and the regular decagon (FM) with the given radius radius r.

In mathematical terms:

ME is the side length of the regular hexagon, the EF of the regular pentagon and a regular decagon of the FM with the given radius radius r.

Occurrence

Nature

The okra fruit has the shape of a pentagon in cross section. The flowers of the morning glory are also formed pentagonal. A variety of cyclic compounds containing a five-membered ring structure (such as cyclopentane, γ -butyrolactone, furan, furanoses etc.).

Architecture and fortifications

Villa Farnese, floor plan

Castle Krzyżtopór

Pamplona in 1845

Satellite image of the Pentagon

The floor plan of a modern bastion fortress has often in the form of a pentagon. Illustrative examples of regular pentagons are, or were, inter alia, the completely rebuilt fortress Bourtange in the Netherlands and Nyenschantz (now St. Petersburg ), the Citadel of Jaca, in the citadel of Pamplona, the fortress Doemitz, the citadel of Turin the citadel of 's- Hertogenbosch, The Citadel of Strasbourg, the citadel of Amiens, the 1598 broken citadel of Vitry -le- François by Girolamo Marini, the lost citadel of Antwerp, which Citadel of Doullens ( Picardy, in part, on regular floor plan ), the Citadel of Lille, the Harburg Castle, the Citadel Vechta, the citadel of Münster, Nieuw- Amsterdam ( Suriname), the castle of Copenhagen, Tilbury Fort in Essex east of London and the height fortress Wülzburg at White Castle in Bavaria. On a regular pentagonal embody the type of fortified palace ( palazzo in Fortezza ) the Villa Farnese, the castles Krzyżtopór and Nowy Wiśnicz and the fortifications of the castle Łańcut in Poland.

Even the Pentagon in Washington uses the regular pentagon as the floor plan. However, it does so not on the old principle of the construction of forts on, but received its form, because it was originally based on another point at which the boundaries of the land pretending this form.

A pentagon is also the facility of the Sanctuary Zelena Hora (Czech Republic) and St. Michael ( Detmold - Hiddesen ) basis.