why does the golden ratio, Φ=√(81/80)+√(1/80)+1/2? I have been trying to understand the mathematics of the golden ratio starting with the following 345 relationship.

Can anybody help me understand why  $ Φ = \sqrt{81/80} + \sqrt{1/80} + 1/2 $ ?

I think it is something to do with these angles but its quite complicated.

I know that $ Φ + 1/Φ = \sqrt{5} $  and  $ Φ - 1/Φ = 1 $.

And also that $ (\sqrt{5} + 2) + (\sqrt{5} - 2) = \sqrt{20} $  and  $ (\sqrt{5} + 2) - (\sqrt{5} - 2) = 4  $.

Which shows a link to the metallic means (1 and 4). If anybody can provide any help it will be greatly appreciated.

As suggested by @Blue:
$$\sqrt{\frac{81}{80}} + \sqrt{\frac{1}{80}}  + \frac12  \ \ \ = \ \ \ \frac{\sqrt{81}}{\sqrt{80}} + \frac{\sqrt{1}}{\sqrt{80}}   + \frac12$$
$$=\frac{9}{\sqrt{80}} + \frac{1}{\sqrt{80}}   + \frac12 \ \ \ = \ \ \ \frac{10}{\sqrt{80}} + \frac12$$
$$=\frac{10}{\sqrt{16 \times 5}} + \frac12 \ \ \ = \ \ \ \frac{2 \times \sqrt{5} \times \sqrt{5}}{4 \sqrt{5}} + \frac12=$$
$$=\frac{\sqrt{5}}{2} + \frac12 \ \ \ = \ \ \ \frac{1+\sqrt{5}}{2} \ \ \ = \ \ \ \Phi$$
