sides = 17-gon
(sides – 2) x 180° = 2700° total angles
2700° total angles ÷ sides = 158.82352941176470588235294117647° perimeter angle
180° – 158.82352941176470588235294117647° perimeter angle = 21.176470588235294117647058823529° radial angle
cos(angle) = adjacent ÷ hypotenuse

radius = 1
perimeter angle ÷ 2 = first cosine angle
158.82352941176470588235294117647° ÷ 2 = 79.411764705882352941176470588235°
cos(79.411764705882352941176470588235) = a ÷ h = a ÷ 1 = a = 0.18374951781657033157440883962073
2 x 0.18374951781657033157440883962073 = 0.36749903563314066314881767924146

0.36749903563314066314881767924146 = shortest diagonal

radial angle ÷ 4 = second cosine angle
21.176470588235294117647058823529° ÷ 4 = 5.2941176470588235294117647058824°
cos(5.2941176470588235294117647058824) = a ÷ h = a ÷ 1 = a = 0.99573417629503452187119117890548
2 x 0.99573417629503452187119117890548 = 1.991468352590069043742382357811

1.991468352590069043742382357811 = longest diagonal = 5.0746921161186659373698293977713

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

longest diagonal ÷ shortest diagonal = 

1.991468352590069043742382357811 ÷ 0.36749903563314066314881767924146 =
= 5.4189757237297103662675177164563

shortest diagonal ÷ longest diagonal = 

0.36749903563314066314881767924146 ÷ 1.991468352590069043742382357811 =
= 0.18453671892660399047930221430902

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

8 x radial angle = 169.41176470588235294117647058823°

180 - 169.41176470588235294117647058823° = 10.588235294117647058823529411768°
10.588235294117647058823529411768° ÷ 2 = 5.294117647058823529411764705884°

radial angle ÷ 4 = 5.294117647058823529411764705884°

perimeter angle ÷ 2 = 79.411764705882352941176470588235°
79.411764705882352941176470588235° + 5.294117647058823529411764705884° = 84.705882352941176470588235294119°