Pentagon

Note: these formulas only work for regular pentagons.

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

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

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

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

R is the radius of the circumcircle.

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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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  The gynoecium of an apple contains five carpels, arranged in a five-pointed star
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  Starfruit is another fruit with fivefold symmetry.

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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.

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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

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