Pentagon
A pentagon is a polygon with five edges. It is defined by five points, which are all on a plane. If all the edges have the same length and the angles at the corners are all 108°, the pentagon is called regular. If the pentagon intersects itself, it is called a pentagram.
Pentagons also occur in nature: Fruits of the Okra are pentangular. The flowers of Ipomoea are pentagular. In chemistry, many Cyclic compounds are pentangles: Cyclopentane and Furan are examples for this. In architecture, many bastions are pentangular: Bourtange, in the Netherlands has been completely restored, and is a pentangle. The Citadel of Lille, Nyenschantz, near St. Petersburg, or the Citadel of Pamplona are . The Villa Farnese is a palace in the form of a pentagon, so is the castle of Nowy Wiśnicz. The Pilgrimage Church of Saint John of Nepomuk near Žďár nad Sázavou also uses a pentangular design.
Formulas
Note: these formulas only work for regular pentagons.
[math]\displaystyle{ \text{Height} = \frac{\sqrt{5+2\sqrt{5}}}{2} \cdot \text{Side}\approx 1.539 \cdot \text{Side}, }[/math]
[math]\displaystyle{ \text{Width} = \text{Diagonal}= \frac{1+\sqrt5}{2} \cdot \text{Side}\approx 1.618 \cdot \text{Side}, }[/math]
[math]\displaystyle{ \text{Width} = \sqrt{2-\frac{2}{\sqrt{5}}} \cdot \text{Height}\approx 1.051 \cdot \text{Height}, }[/math]
[math]\displaystyle{ \text{Diagonal} = R\ {\sqrt { \frac {5+\sqrt{5}}{2}} } = 2R\cos 18^\circ = 2R\cos\frac{\pi}{10} \approx 1.902 R, }[/math]
R is the radius of the circumcircle.
Examples of pentagons
Plants
Pentagonal cross-section of okra.
Morning glories, like many other flowers, have a pentagonal shape.
The gynoecium of an apple contains five carpels, arranged in a five-pointed star
Starfruit is another fruit with fivefold symmetry.
Animals
A sea star. Many echinoderms have fivefold radial symmetry.
An illustration of brittle stars, also echinoderms with a pentagonal shape.
Artificial
The Pentagon, headquarters of the United States Department of Defense.
Home plate of a baseball field
Fort Bourtange,in the Netherlands
