User:Frederic.wang/sandbox

Derivation of the diagonal length formula
The diagonals of a regular pentagon (here represented by d) can be calculated based upon the golden ratio $$\varphi$$ and the known side t: $$\frac {d}{t} = \varphi = \frac {1+ \sqrt {5} }{2} \, $$

Accordingly:

$$d = t \times \varphi \ = R\ {\sqrt { \frac {5+\sqrt{5}}{2}} } = 2R\cos 18^\circ = 2R\cos\frac{\pi}{10} \approx 1.90211303259 R.$$