Pentagon

2015-01-02
Pentagon Dimensions

Pentagon Dimensions

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 = sin 36 = \(\sqrt{\frac{5-\sqrt5}{8}}\) FJ=JK/Φ
AG= 2cos 54 = 2sin 36= \(\sqrt{\frac{5-\sqrt5}{2}}\) GJ=AJ/Φ
CH=DH=BK=CK=1/2 GH=AF/Φ=\(\frac12\sqrt{5-2\sqrt5}\)

 

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