sides = 5 (pentagon)
(sides – 2) x 180° = 540° total angles
540° total angles ÷ sides = 108° perimeter angle
180° – 108° perimeter angle = 72° radial angle
cos(angle) = adjacent ÷ hypotenuse

radius = 1
perimeter angle ÷ 2 = first cosine angle
108° ÷ 2 = 54°
cos(54) = a ÷ h = a ÷ 1 = a = 0.58778525229247312916870595463907
2 x 0.58778525229247312916870595463907 = 1.1755705045849462583374119092781

1.1755705045849462583374119092781 = shortest diagonal

1.1755705045849462583374119092781^2 = 1.3819660112501051517954131656344
1.3819660112501051517954131656344 – 1 = 0.3819660112501051517954131656344
√0.3819660112501051517954131656344 = 0.6180339887498948482045868343656

radial angle ÷ 4 = second cosine angle
72° ÷ 4 = 18°
cos(18) = a ÷ h = a ÷ 1 = a = 0.95105651629515357211643933337938
2 x 0.95105651629515357211643933337938 = 1.9021130325903071442328786667588

1.9021130325903071442328786667588 = longest diagonal

1.9021130325903071442328786667588^2 = 3.6180339887498948482045868343656
3.6180339887498948482045868343656 – 2 = 1.6180339887498948482045868343656

============================================================

longest diagonal ÷ shortest diagonal = 

1.9021130325903071442328786667588 ÷ 1.1755705045849462583374119092781 =
= 1.6180339887498948482045868343657

shortest diagonal ÷ longest diagonal = 

1.1755705045849462583374119092781 ÷ 1.9021130325903071442328786667588 =
= 0.6180339887498948482045868343657