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.

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

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

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

$Diagonal = R \sqrt{\frac{5 + \sqrt{5}}{2}} = 2 R \cos 1 8^{\circ} = 2 R \cos \frac{π}{10} \approx 1.902 R,$

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.