What is the ratio of side lengths of Cyclic regular Pentagon and a circumscribed regular pentagon?

Question

What is the ratio of side lengths of a cyclic regular pentagon and a circumscribed regular pentagon ? I’ve tried using similar triangle but there are too many unknown number. Or should I use golden ratio or Trigonometric function? Any help or hint is appropriated.

Answer 1

From similar triangles OAB andf OCD, it is the ratio of the distances from the centers to the sides, i.e.

$$\frac{OB}{OD} = \frac{OB}{OA} = \cos36^\circ$$

Answer 2

Let $DO=r$.

Thus, $$DC=r\tan36^{\circ}=\frac{r\sqrt{1-\cos^236^{\circ}}}{\cos36^{\circ}}=\frac{r\sqrt{1-\left(\frac{1+\sqrt5}{4}\right)^2}}{(\frac{\sqrt5+1}{4})}=...$$ $$AB=r\sin54^{\circ}=\frac{r(1+\sqrt5)}{4}.$$ Can you end it now?

Answer 3

$\dfrac{CD}{DO}=\tan{36°}$

$CD=r\tan{36°}$

$\dfrac{BA}{AO}=\cos{36°}$

$BA=r\cos{36°}$

$\dfrac{BA}{CD}=\dfrac{r\cos{36°}}{r\tan{36°}}=\dfrac{\cos{36°}}{\dfrac{\sin{36°}}{\cos{36°}}}$

$\text{Ratio}=1:\sin{36°}$

Whatever that is. There are ways to evaluate it but I think that's a suitable answer.