Pentagon

A regular pentagon has the following dimensions:

AB=BC=CD=DE=AE=BG=EG=1

AC=AD=BD=BE=CE=Φ

BF=EF=Φ/2 = cos 36 = sin 54 = $\frac{1+\sqrt{5}}{4}$

CG=DG= Φ-1 = 1/Φ = $\frac{\sqrt{5}-1}{2}$

AF=FG= cos 54 = sin 36 = $\sqrt{\frac{5-\sqrt{5}}{8}}$

AG= 2cos 54 = 2sin 36= $\sqrt{\frac{5-\sqrt{5}}{2}}$

CH=DH=BK=CK=1/2

Height AH=EK=$\frac{1}{2}\cos 18=\frac{1}{2}\sin 72=\frac{1}{2}\sqrt{5+2\sqrt{5}}$

Circumradius AJ=$\sqrt{\frac{5+\sqrt{5}}{10}}$ = BJ=CJ=DJ=EJ

Inradius JK=HJ= $\sqrt{\frac{5+2\sqrt{5}}{20}}$ = $\frac{AH}{\sqrt{5}}$

FH= cos 18 =sin 72 = $\sqrt{\frac{5+\sqrt{5}}{8}}$

FJ=JK/Φ

GJ=AJ/Φ

GH=AF/Φ=$\frac{1}{2}\sqrt{5-2\sqrt{5}}$

ABGE is a equilateral rhombus, with angles {108°, 54°, 108°, 54°} and diagonals of length Φ and $\sqrt{\frac{5-\sqrt{5}}{2}}$.

The small pentagon in the center has sides $1/{\mathrm{\Phi }}^2=\frac{3-\sqrt{5}}{2}$ times that of the larger.