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[change]
Note: these formulas only work for regular pentagons.
<math>\text{Height} = \frac{\sqrt{5+2\sqrt{5}}}{2} \cdot \text{Side}\approx 1.539 \cdot \text{Side},</math>
<math>\text{Width} = \text{Diagonal}= \frac{1+\sqrt5}{2} \cdot \text{Side}\approx 1.618 \cdot \text{Side},</math>
<math>\text{Width} = \sqrt{2-\frac{2}{\sqrt{5}}} \cdot \text{Height}\approx 1.051 \cdot \text{Height},</math>
<math>\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[change]
Plants[change]
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Pentagonal cross-section of okra.
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Morning glories, like many other flowers, have a pentagonal shape.
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Starfruit is another fruit with fivefold symmetry.
Animals[change]
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A sea star. Many echinoderms have fivefold radial symmetry.
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An illustration of brittle stars, also echinoderms with a pentagonal shape.
Artificial[change]
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The Pentagon, headquarters of the United States Department of Defense.
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Home plate of a baseball field
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Fort Bourtange,in the Netherlands
