Regular pentagon  

A regular pentagon 

Edges and vertices  5 
Schläfli symbol  {5} 
Coxeter–Dynkin diagram  
Symmetry group  Dihedral (D_{5}) 
Internal angle (degrees) 
108° 
Properties  convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a pentagon is any fivesided polygon. A pentagon may be simple or selfintersecting. The internal angles in a simple pentagon total 540°. A pentagram is an example of a selfintersecting pentagon. It is a 2D shape.The 3D shape is called a pentagonal prism.
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A regular pentagon has all sides of equal length and all interior angles are equal measure (108°). It has five lines of reflectional symmetry and it has rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). Its Schläfli symbol is {5}. The chords of a regular pentagon are in golden ratio to its sides.
The area of a regular convex pentagon with side length t is given by
A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression
The area of any regular polygon is:
where P is the perimeter of the polygon, and a is the apothem. One can then substitute the respective values for P and a, which makes the formula:
with t as the given side length. Then we can then rearrange the formula as:
and then, we combine the two terms to get the final formula, which is:
The diagonals of a regular pentagon (hereby represented by D) can be calculated using the following formula:
where T = the side length of the pentagon, itself.
A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.
One method to construct a regular pentagon in a given circle is as follows:
An alternative method is this:
A direct method using degrees follows:
After forming a regular convex pentagon, if you join the nonadjacent corners (drawing the diagonals of the pentagon), you obtain a pentagram, with a smaller regular pentagon in the center. Or if you extend the sides until the nonadjacent ones meet, you obtain a larger pentagram.
A simple method of creating a regular pentagon from just a strip of paper is by tying an overhand knot into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a pentagram when backlit.
Pentagonal crosssection of okra. 
Morning glories, like many other flowers, have a pentagonal shape. 
The gynoecium of an apple contains five carpels, arranged in a fivepointed star 
Starfruit is another fruit with fivefold symmetry. 
A sea star. Many echinoderms have fivefold radial symmetry. 
An illustration of brittle stars, also echinoderms with a pentagonal shape. 
A pentagon cannot appear in any tiling made by regular polygons. To prove a pentagon cannot form a regular tiling, 360 / 108 = 3 1/3, which is not a whole number. More difficult is proving a pentagon cannot be in any tiling made by regular polygons:
There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126 degrees. To find the number of sides this polygon has, the result is 360 / (180 − 126) = 6 2/3, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.

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Singular 
Plural 
Pentagon
Wikipedia ^{de}
Pentagon n.
This German entry was created from the translations listed at pentagon. It may be less reliable than other entries, and may be missing parts of speech or additional senses. Please also see Pentagon in the German Wiktionary. This notice will be removed when the entry is checked. (more information) April 2008
[[File:thumbrightA pentagon]] A pentagon is a polygon with 5 edges.
