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.

${\text{Height}}={\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\cdot {\text{Side}}\approx 1.539\cdot {\text{Side}},$

${\text{Width}}={\text{Diagonal}}={\frac {1+{\sqrt {5}}}{2}}\cdot {\text{Side}}\approx 1.618\cdot {\text{Side}},$

${\text{Width}}={\sqrt {2-{\frac {2}{\sqrt {5}}}}}\cdot {\text{Height}}\approx 1.051\cdot {\text{Height}},$

${\text{Diagonal}}=R\ {\sqrt {\frac {5+{\sqrt {5}}}{2}}}=2R\cos 18^{\circ }=2R\cos {\frac {\pi }{10}}\approx 1.902R,$

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.