A regular pentagon has the following dimensions: AB=BC=CD=DE=AE=BG=EG=1 Height AH=EK=\(\frac12\cos 18 = \frac12\sin 72= \frac12\sqrt{5+2\sqrt5}\) AC=AD=BD=BE=CE=Φ Circumcenter AJ=\(\sqrt{\frac{5+\sqrt5}{10}}\)=BJ=CJ=DJ=EJ BF=EF=Φ/2 = cos 36 = sin 54 = \(\frac{1+\sqrt5}{4}\) Inradius JK=HJ=\(\sqrt{\frac{5+2\sqrt5}{20}}\)=\(\frac{AH}{\sqrt5}\) CG=DG= Φ-1 = 1/Φ = \(\frac{\sqrt5-1}{2}\) FH= cos 18 =sin 72 = \(\sqrt{\frac{5+\sqrt5}{8}}\) AF=FG= cos 54 **…Read the Rest**

## Geometry

### Phi, the Golden Ratio

2011-10-23

Phi \((\Phi, \phi)\) is a Greek letter that mathematicians have assigned to a specific ratio or proportion, called the golden ratio, that most people find to be attractive in art, architecture, and nature. The golden ratio is illustrated as \(\frac{A}{B}=\frac{A+B}{A}\equiv\phi\). The only positive solution is **…Read the Rest**